AND NO ONE TAKES THE HONOR TO HIMSELF: kai ouch heauto tis lambanei (3SPAI) ten timen: (Exodus 28:1; Leviticus 8:2; Numbers 3:3; 16:5,7,10,35,40,46-48; 17:3-11; 18:1-5; 1 Chronicles 23:13; 2 Chronicles 26:18; John 3:27)

The writer continues with the second requirement of the Jewish high priest, indicating that he must be chosen and appointed by God.

No one (ouch) indicates absolute negation. 

This same Greek word (time) is used by the Jewish Historian Josephus to denote the “office” of high priest (Ant. 3.188–89).

BUT RECEIVES IT WHEN HE IS CALLED BY GOD EVEN AS AARON WAS: alla kaloumenos (PPPMSN) hupo tou theou kathosper kai Aaron:

But receives it (honor) - Receives is not in the Greek text but is supplied by  the translators to lend fluidity to the sentence.

John the Baptist declared...

"A man can receive nothing, unless it has been given him from heaven." (John 3:27)

The office of high priest could not be campaigned for but was given by right of birth to the man chosen by God. It was an honor no man could take to himself.

When he is called - This refers to a call to a particular position and service to God. In the Old Testament this call to the Levitical priesthood was only given to certain men...

"Then bring near to yourself Aaron your brother, and his sons with him, from among the sons of Israel, to minister as priest to Me-- Aaron, Nadab and Abihu, Eleazar and Ithamar, Aaron's sons. (Exodus 28:1)

"Take Aaron and his sons with him, and the garments and the anointing oil and the bull of the sin offering, and the two rams and the basket of unleavened bread (Leviticus 8:2)

The sons of Amram were Aaron and Moses. And Aaron was set apart to sanctify him as most holy, he and his sons forever, to burn incense before the LORD, to minister to Him and to bless in His name forever. (1 Chronicles 23:13)

There are some dreadful instances where men presumed to act as priests who were not priests, such as Korah (Numbers 16), Saul (1 Samuel 13) and King Uzziah who disregarded the fact that no one was to attempt to function as a priest on his own volition. Scripture records that when King Uzziah...

became strong, his heart was so proud that he acted corruptly, and he was unfaithful to the LORD his God, for he entered the temple of the LORD to burn incense on the altar of incense...18 And they opposed Uzziah the king and said to him, "It is not for you, Uzziah, to burn incense to the LORD, but for the priests, the sons of Aaron who are consecrated to burn incense. Get out of the sanctuary, for you have been unfaithful, and will have no honor from the LORD God." 19 But Uzziah, with a censer in his hand for burning incense, was enraged; and while he was enraged with the priests, the leprosy broke out on his forehead before the priests in the house of the LORD, beside the altar of incense. 20 And Azariah the chief priest and all the priests looked at him, and behold, he was leprous on his forehead; and they hurried him out of there, and he himself also hastened to get out because the LORD had smitten him.  (2 Chronicles 26:16, 18-20)

In the OT a call of God to undeserving men was part of His gracious dealing with sinful mankind. We see examples of God's graciously calling individuals to certain "ministries"...

Abraham

1 Now the LORD said to Abram, "Go forth from your country, And from your relatives And from your father's house, To the land which I will show you;
2 And I will make you a great nation, And I will bless you, And make your name great; And so you shall be a blessing;
3 And I will bless those who bless you, And the one who curses you I will curse. And in you all the families of the earth shall be blessed." (Ge 12:1-3)

Moses

"Therefore, come now, and I will send you to Pharaoh, so that you may bring My people, the sons of Israel, out of Egypt." (Ex 3:10)

 Bezaleel

2 "See, I have called by name Bezalel, the son of Uri, the son of Hur, of the tribe of Judah. 3 "And I have filled him with the Spirit of God in wisdom, in understanding, in knowledge, and in all kinds of craftsmanship, 4 to make artistic designs for work in gold, in silver, and in bronze, 5 and in the cutting of stones for settings, and in the carving of wood, that he may work in all kinds of craftsmanship. (Ex 31:2-5)

Isaiah

8 Then I heard the voice of the Lord, saying, "Whom shall I send, and who will go for Us?" Then I said, "Here am I. Send me!"9 And He said, "Go, and tell this people: 'Keep on listening, but do not perceive; Keep on looking, but do not understand. (Isaiah 6:8, 9)

Jeremiah

Now the word of the LORD came to me saying, 5 "Before I formed you in the womb I knew you, And before you were born I consecrated you; I have appointed you a prophet to the nations." (Jer 1:4, 5),

Ezekiel

Then He said to me, "Son of man, I am sending you to the sons of Israel, to a rebellious people who have rebelled against Me; they and their fathers have transgressed against Me to this very day. 4 "And I am sending you to them who are stubborn and obstinate children; and you shall say to them, 'Thus says the Lord GOD.' (Ezek 2:3-4)

All New Testament believers have been called into God's priesthood as Peter declared in his first epistle writing that...

you are A CHOSEN RACE, A royal PRIESTHOOD, A HOLY NATION, A PEOPLE FOR God's OWN POSSESSION, that you may proclaim the excellencies of Him who has called you out of darkness into His marvelous light (see note 1 Peter 2:9) (Note: Do not misinterpret Peter for he is not saying that the NT church replaces Israel in God's prophetic timetable - see discussion of Israel of God)

The question we each to ask is "Am I fulfilling my calling by God to the priesthood of believers?"

Guzik  qualifies the believers' priesthood noting that...

We can also not take the honor of being our own priest. It is great arrogance to think we can approach God on our own, without a priest; but it is great superstition to think we need any other priest other than Jesus Christ Himself. God has provided a mediator, a priest, and we must avail ourselves of the priest God has provided.

Matthew Poole in his excellent mid-seventeenth century commentary wrote that...

A sinner can undertake to manage nothing towards God immediately, or by himself, but with a mediating priest, who must know God’s mind and perform it . . . The common sense of mankind about it since the fall doth evidence it; no nation being without a religion, a temple, a place of worship, or a priest.

C. H. Spurgeon had the following assessment regarding Matthew Poole's commentary writing that...

If I must have only one commentary, and had read Matthew Henry as I have, I do not know but what I should choose Poole. He is a very prudent and judicious commentator... not so pithy and witty by far as Matthew Henry, but he is perhaps more accurate, less a commentator, and more an expositor.

SO ALSO CHRIST DID NOT GLORIFY HIMSELF SO AS TO BECOME A HIGH PRIEST: Houtos kai o Christos ouch heauton edoxasen (3SAAI) genethenai (APN) archierea: (John 7:18; 8:54) (Hebrews 1:5; Psalms 2:7; Micah 5:2; John 3:16; Acts 13:33; Romans 8:3)

Having just stated that no man was entitled to appoint himself as high priest but that he became such only by divine call, even Christ did not make Himself High Priest, but God the Father recognized Him as such. The writer's logic is that just as much as Jesus was declared to be the Son of God by God Himself (Psalm 2:7), so also He was declared to be a priest forever according to a different order (Psalm 110:4).

The primary ministry of Jesus today is as our forever High Priest Who represents us before God because Who chose His Son to be our eternal Priest. If we were first century Jews who had believed in the Messiah, it would be difficult to grasp the truth that Jesus was the Great High Priest. He was not from the line of Aaron. Even when He was in the Temple in Jerusalem, He did not attempt to practice what the Jews would recognize as a priestly ministry. And so the writer goes into a lengthy explanation (which he picks up again in chapter 7) about how Jesus would be qualified to be a High Priest. The writer is trying to get his Jewish readers to make what we would call today a "paradigm shift" in their thinking.

And so Jesus did not glorify Himself. Instead in John Jesus speaks of the One Who gives glory...

John 7:18 "He who speaks from himself seeks his own glory; but He who is seeking the glory of the One Who sent Him (God the Father), He is true, and there is no unrighteousness in Him.

John 8:54 Jesus answered, "If I glorify Myself, My glory is nothing; it is My Father who glorifies Me, of Whom you say, 'He is our God';

High priest (749) (archiereus from arche = first in a series, the leader or ruler + hiereus = priest) (Dictionary articles - Easton's; ISBE) refers to the priest that was chief over all the other priests in Israel. This office was established by God through Moses instructions in the Pentateuch. The high priest functioned as the mediator between Jehovah and Israel performing sacrifices and rituals like other priests, but in addition acting to expiate the sins of the nation on the annual Day of Atonement.

The irony is that the high priest Caiaphas was residing over the Sanhedrin during trial of Jesus, the trial which would lead to His death and pave the way for His eternal High Priesthood!

Eerdman's Bible Dictionary explains that...

The high priest descended from Eleazar, the son of Aaron. The office was normally hereditary and was conferred upon an individual for life (Nu 25:10-13). The candidate was consecrated in a seven-day ceremony which included investiture with the special clothing of his office as well as anointments and sacrifices (Ex 29:1-37; Lev 8:5-35).

The high priest was bound to a higher degree of ritual purity than ordinary Levitical priests. He could have no contact with dead bodies, including those of his parents. Nor could he rend his clothing or allow his hair to grow out as signs of mourning. He could not marry a widow, divorced woman, or harlot, but only an Israelite virgin (Lev. 21:10-15). Any sin committed by the high priest brought guilt upon the entire nation and had to be countered by special sacrifice (Lev 4:1-12). Upon a high priest’s death manslayers were released from the cities of refuge (Nu 35:25, 28, 32).  (Eerdman's Bible Dictionary)

Archiereus occurs only in the Gospels (Matthew - 25 times, Mark 21 times, Luke 15 times, John 20 times), Acts 22 times and 17 times in Hebrews. The references to the high priests in the Gospels and Acts refers primarily to their bitter opposition to Jesus Who the writer of Hebrews identifies as our everlasting High Priest.

BUT HE WHO SAID TO HIM THOU ART MY SON, TODAY I HAVE BEGOTTEN THEE: all o lalesas (AAPMSN) pros auton Huios mou ei (2SPAI) su ego semeron gegenneka (1SRAI) se:

Psalm 2 is one Psalm which is indisputably Messianic as shown by the fact that the NT writers quote from it in references that are clearly speaking of Jesus. The reader is encouraged to listen to the two part exposition (each about 45 minutes) of this great psalm by my dear brother in Christ, Tony Garland - Mp3 Part 1 ; Mp3 Part 2 (Or see the Pdf transcript Why Do the Nations Rage?). The Jewish readers of this exhortational letter ("sermon") to the Hebrews undoubtedly understood Psalm 2 and Psalm 110 (quoted in the next verse  - Hebrews 5:6) as prophecies related to the Messiah, their Hope. And so the writer buttresses his argument by quoting from Psalm 2:7.  As you recall, the writer had previously quoted from Psalm 2:7 in Hebrews 1:5 (note) using this quotation to establish the fact that Christ was superior to the angels. Now in Hebrews 5 the writer quotes from this same psalm to help him establish that the priesthood of Christ is better than the Levitical priesthood.

Psalm 2:7 "I will surely tell of the decree of the LORD: He said to Me, 'Thou art My Son, Today I have begotten Thee.

Psalm 2:7 is also quoted by Paul in Acts 13. Upon Jesus' resurrection God is said to have declared Christ as begotten, Luke recording Paul's quotation from Psalm 2 stating...

that God has fulfilled this promise to our children in that He raised up Jesus, as it is also written in the second Psalm, 'THOU ART MY SON; TODAY I HAVE BEGOTTEN THEE.' (Acts 13:3)

In the context of Acts 13 Paul  relates the prophecy in Psalm 2:7 to Christ's resurrection, rather than His incarnation.  Jesus’ resurrection from the dead marks the time at which Jesus could fully assume His role as our great High Priest. In other words, prior to this time (as far as I can discern from the Scriptures) Jesus did not function specifically as the Great High Priest.

Note that in Acts, Paul referred to Psalm 2 as the "the second psalm," supporting that the chapter divisions in the book of Psalms are not the product of medieval scholars.

In a parallel passage Paul writes that Jesus...

was declared the Son of God with power by the resurrection from the dead (i.e., the resurrection of Jesus is proof of His deity), according to the Spirit of holiness, Jesus Christ our Lord (see note Romans 1:4)

Spurgeon commenting on Psalm 2:7 writes...

This Psalm wears something of a dramatic form, for now another person is introduced as speaking. We have looked into the council chamber of the wicked, and to the throne of God, and now we behold the Anointed (Ed note: the Messiah) declaring His rights of sovereignty, and warning the traitors of their doom.

God has laughed at the counsel and ravings of the wicked, and now Christ, the Anointed, Himself comes forward, as the Risen Redeemer, "declared to be the Son of God with power, according to the spirit of holiness, by the resurrection from the dead." Romans 1:4. Looking into the angry faces of the rebellious kings, the Anointed One seems to say, "If this sufficeth not to make you silent."

I will declare the decree. Now this decree is directly in conflict with the device of man, for its tenor is the establishment of the very dominion against which the nations are raving.

Thou art my Son. Here is a noble proof of the glorious Divinity of our Immanuel. "For unto which of the angels said he at any time, Thou art my Son, this day have I begotten thee?" What a mercy to have a Divine Redeemer in Whom to rest our confidence!

This day have I begotten thee. If this refers to the Godhead of our Lord, let us not attempt to fathom it, for it is a great truth, a truth reverently to be received, but not irreverently to be scanned. It may be added, that if this relates to the Begotten One in his human nature, we must here also rejoice in the mystery, but not attempt to violate its sanctity by intrusive prying into the secrets of the Eternal God. The things which are revealed are enough, without venturing into vain speculations. In attempting to define the Trinity, or unveil the essence of Divinity, many men have lost themselves: here great ships have foundered. What have we to do in such a sea with our frail skiffs?

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Octavius Winslow has the following devotional on Hebrews 5:5 - The Atonement of Christ is of infinite value and efficacy. If Christ were a mere creature, if He claimed no higher dignity than Gabriel, or one of the prophets or apostles, then His atonement, as it regards the satisfaction of Divine justice, the honoring of the law, the pardon of sin, the peace of the conscience, and the salvation of the soul, would possess no intrinsic efficacy whatever. It would be but the atonement of a finite being - a being that possessing no superior merit to those in whose behalf the atonement was made. We state it, then, broadly and unequivocally, that the entire glory, dignity, value, and efficacy of Christ's precious blood which He shed for sin rests entirely upon the Deity of His person. If the Deity of Christ sinks, the atonement of Christ sinks with it; if the one stands, so stands the other. How strong are the words of Paul, addressed to the Ephesian elders: "Take heed therefore unto yourselves, and to all the flock over which the Holy Spirit has made you overseers, to feed the church of God which He has purchased with His own blood." How conclusive is this testimony! The blood that purchased the church was Divine. It was indeed the blood of Christ's humanity-for His human nature alone could suffer, bleed, and die-yet deriving all its glory, value, and efficacy from the union of the human with the Divine nature. It was the blood of the God-man, Jehovah Jesus-no inferior blood could have sufficed. The law which Adam, our federal head, broke, before it could release the sinner from its penalty, demanded a sacrifice infinitely holy, and infinitely great: one equal with the Father-the dignity of whose person would impart infinite merit to His work, and the infinite merit of whose work would fully sustain its honor and its purity. All this was found in the person of Christ. In His complex person He was eminently fitted for the mighty work. As God, He obeyed the precepts and maintained the honor of the law; as man, He bore its curse and endured its penalty. It was the blending as into one these two natures; the bringing together these extremes of being, the finite and the infinite, which shed such resplendent luster on His atonement, which stamped such worth and efficacy on His blood. Dear Reader, treat not this subject lightly, deem it not a useless speculation! It is of the deepest moment. If the blood of Christ possesses not infinite merit, infinite worth, it could never be efficacious in washing away the guilt of sin, or in removing the dread of condemnation. When you come to die, this, of all truths, if you are an experimental believer, will be the most precious and sustaining. In that solemn hour, when the curtain that conceals the future parts, and eternity lets down upon the view the full blaze of its awful realities-in that hour, when all false dependencies will crumble beneath you, and sin's long catalogue passes in review before you-oh, then to know that the Savior on whom you depend is God in your nature-that the blood in which you have washed has in it all the efficacy and value of Deity-this, this will be the alone plank that will buoy up the soul in that awful moment, and at that fearful crisis. Oh precious truth this, for a poor believing soul to rest upon! We wonder not that, fast anchored on this truth, amid circumstances the most appalling, death in view; wearing even its most terrific aspect, the believer in Jesus can survey the scene with composure, and quietly yield his spirit into the hands of Him who redeemed it.

JUST AS HE SAYS ALSO IN ANOTHER PASSAGE: kathos kai en hetero legei (3SPAI): (Heb 5:10; 6:20; 7:3,15,17,21; Psalm 110:4) (Genesis 14:18,19)

In another place - The Greek is heteros referring to another but one that is different. To illustrate in Acts 7:18 Luke records that after Joseph's death "another (heteros) king (pharaoh) arose" one of quite a different character.

The Priesthood of our Lord was not according to the Levitical order but was of a completely different character, an order that had been prophesied almost 1000 years earlier in the Messianic psalm 110:4.

THOU ART A PRIEST FOREVER ACCORDING TO THE ORDER OF MELCHIZEDEK: Su hiereus eis ton aiona kata ten taxin Melchisedek: (Heb 5:10; 6:20; 7:17,21; Psalm 110:4)

Recall the writer discusses three requirements for the priesthood in Hebrews 5...

(1) He was appointed on men’s behalf to deal with the things concerning God and as such functioned as the link between God and man.

(2) The priest must be one with men and must have gone through men’s experiences so that his sympathy would be with them.

(3) No man appoints himself to the priesthood but his appointment is of God. The priesthood is not a coveted office to be taken but a glorious privilege to which one is called. This latter point is applicable to all believers today, for all have been made priests in Christ Jesus. It follows that their specific ministry to and for God should not be one they chose, but one they received from God.

Christ met all requirements for the priesthood except that of descent from Aaron. And yet this seeming deficiency did not disqualify Him, for He was of another priestly order, greater even than that of Aaron and Levi. The psalmist had prophesied of this greater order...

Ps 110:4 The LORD has sworn and will not change His mind, "Thou art a priest forever According to the order of Melchizedek."

Psalm 110:4 is quoted four times in Hebrews, here in Hebrews 5:6 and...

Hebrews 5:10 (note)  being designated by God as a high priest according to the order of Melchizedek.

Hebrews 6:20 (note) where Jesus has entered as a forerunner for us, having become a high priest forever according to the order of Melchizedek.

Hebrews 7:17 (note) For it is witnessed of Him, "THOU ART A PRIEST FOREVER ACCORDING TO THE ORDER OF MELCHIZEDEK."

Hebrews 7:21 (note) (for they indeed became priests without an oath, but He with an oath through the One who said to Him, "THE LORD HAS SWORN AND WILL NOT CHANGE HIS MIND, 'THOU ART A PRIEST FOREVER'")

Melchizedek was the mysterious priest/king who met Abraham when he returned from defeating the marauding confederation of kings from the north (see Genesis 14:17-19). In the Genesis account, he is merely called "king of Salem" (meaning "peace") and "priest of the most high God" (the Hebrew name is El Elyon, "highest God") (El Elyon: Most High God - Sovereign Over All).

Melchizedek is discussed in greater detail in Hebrews 7:1-21 and as Harry Ironside writes...

It is enough to point out here that Melchizedek was recognized as priest of the most high God centuries before the Levitical priesthood came into existence. The Levitical priesthood and the legal covenant with which it was connected had their place until the Son, who was to fulfill the Melchizedekian type, should come. (H. A. Ironside Expository Commentary on Hebrews)

Scofield comments that...

Melchizedek was a suitable type of Christ as High Priest, because: (1) he was a man (Heb 7:4 1Ti 2:5); (2) he was a king-priest (Ge 14:18 w Zec 6:12,13); (3) his name means "my king is righteous" (Isa 11:5), and he was king of Salem (peace cp. Isa 11:6-9); (4) he had no recorded "beginning of days" (cp. J 1:1) or "end of life" (cp. Ro 6:9; Heb 7:23-25), nor (5) was he made a high priest by human appointment (Ps 110:4). But the contrast between the high priesthood of Melchizedek and Aaron is only as to person, "order" (or appointment), and duration. In His work Christ follows the Aaronic pattern, the "shadow" of which Christ was the substance (Heb 8:1-6; 9:1-28).

C H Spurgeon has a lengthy comment on Psalm 110:4 writing that...

We have now reached the heart of the psalm, which is also the very centre and soul of our faith. Our Lord Jesus is a Priest King by the ancient oath of Jehovah: "He glorified not Himself to be made a High Priest," but was ordained there unto from of old, and was called of God a High Priest after the order of Melchizedek.

It must be a solemn and a sure matter which leads the Eternal (God) to swear, and with Him an oath fixes and settles the decree for ever; but in this case, as if to make assurance a thousand times sure, it is added," and will not repent."

It is done, and done for ever and ever; Jesus is sworn in to be the Priest of His people, and He must abide so even to the end, because His commission is sealed by the unchanging oath of the immutable Jehovah. If His priesthood could be revoked, and His authority removed, it would be the end of all hope and life for the people whom He loves; but this sure rock is the basis of our security -- the oath of God establishes our glorious Lord both in His priesthood and in His throne. It is the Lord Who has constituted Him a priest for ever, He has done it by oath, that oath is without repentance, is taking effect now, and will stand throughout all ages: hence our security in Him is placed beyond all question.

The declaration runs in the present tense as being the only time with the Lord, and comprehending all other times. "Thou art," i.e., Thou wast and art and art to come, in all ages a priestly King.

The order of Melchizedek's priesthood was the most ancient and primitive, the most free from ritual and ceremony, the most natural and simple, and at the same time the most honourable. That ancient patriarch was the father of his people, and at the same time ruled and taught them; he swayed both the sceptre and the censer, reigned in righteousness, and offered sacrifice before the Lord. There has never arisen another like to him since his days, for whenever the kings of Judah attempted to seize the sacerdotal office they were driven back to their confusion: God would have no king priest save His son. Melchizedek's office was exceptional -- none preceded or succeeded him; he comes upon the page of history mysteriously; no pedigree is given, no date of birth, or mention of death; he blesses Abraham, receives tithe and vanishes from the scene amid honours which show that he was greater than the founder of the chosen nation. He is seen but once, and that once suffices.

Aaron and his seed came and went; their imperfect sacrifice continued for many generations, because it had no finality in it, and could never make the comers thereunto perfect.

Our Lord Jesus, like Melchizedek, stands forth before us as a Priest of divine ordaining; not made a priest by fleshly birth, as the sons of Aaron: He mentions neither father, mother, nor descent, as His right to the sacred office; He stands upon His personal merits, by Himself alone; as no man came before Him in his work, so none can follow after; His order begins and ends in His own person, and in Himself it is eternal,

"having neither beginning of days nor end of years The King Priest has been here and left his blessing upon the believing, and now he sits in glory in his complete character, stoning for us by the merit of his blood, and exercising all power on our behalf."

"O may we ever hear thy voice
In mercy to us speak,
And in our Priest we will rejoice,
Thou great Melchizedek."

The last verses of this psalm we understand to refer to the future victories of the Priest King. He shall not forever sit in waiting posture, but shall come into the fight to end the weary war by His own victorious presence. He will lead the final charge in person; His own right hand and His holy arm shall get unto Him the victory.