Men (3748) (hostis) is more literally "The very ones who" including ( Hymenaeus and Philetus).

Wuest notes that hostis and hoítines have...

the two-fold function of pointing out and showing character. It refers back to Hymenaeus and Philetus, “the very ones who are of such a character as to.” The word, Expositors says, “implies that Hymenaeus and Philetus were only the more conspicuous members of a class of false teachers.” (Wuest, K. S. Wuest's Word Studies from the Greek New Testament: Eerdmans) (Bolding added)

Gone astray (795) (astocheo from ástochos = one who misses his aim <> in turn from stóchos = aim, target <> which is from a = negative particle + stoichos (an aim) or stochos - mark) means to miss the mark, deviate from truth: swerve. To err, deviate in a figurative and spiritual sense.

Literally Paul is saying these men

concerning the truth, missed the mark or deviated from the truth.

Astocheo does not mean to miss achieving the aim that one has set, but not to set the proper aim at which one ought to aim. It is not focusing on the right goal instead of not achieving one’s set aim. Naturally if one specializes in the proclamation of something that is not essential and central, he will inevitably neglect that which is central and important. The verb does not mean what is conceived by some as “to fall from grace.”

The false teachers and their followers had clearly demonstrated by their deviation from Apostolic truth that they had left the straight path of sound doctrine. They had deviated from "The Way" an early title for the church. They had deviated from "the Way, the Truth and the Life" (John 14:6) and the result was upheaval of faith for faith comes by hearing (truth) and hearing (truth) equates with the pure milk of the Word of God.

So Paul is saying here are some men who have embraced TRUTH for a while and then gone astray from that TRUTH. This emphasizes Paul's charge for us to be very "diligent...(to handle) accurately the Word of Truth" for clearly it can be mishandled with disastrous consequences.

Note that astocheo is used only by Paul in the NT,  the other two uses being found in the first epistle to Timothy...

For some men, straying from these things (What things? verse 5 ''Instruction'' which is fruitful), have turned aside (astocheo) to fruitless (profitless for their talk produced no godliness. Shakespeare described such babble as “full of sound and fury, signifying nothing.” God said it was ''straw...(not) fire'' in Jer 23:28) discussion (It wasn't that they were missing what they had aimed at, for this verb indicates that their problem was they had not set the proper aim, i.e., instruction which yields spiritual fruit), 7 wanting to be teachers of the Law, even though they do not understand either what they are saying or the matters about which they make confident assertions. (1Timothy 1:7-8)

O Timothy, guard (see word study on phulasso = aorist imperative) what has been entrusted (placed beside you as a deposit consigned to Timothy for faithful keeping and faith teaching to faithful men who would be able to teach others also) to you, avoiding worldly and empty chatter and the opposing arguments of what is falsely called "knowledge"--21 which some have professed (announced with certainty, asserted respecting themselves) and thus gone astray (astocheo) from "the faith (pistis). Grace be with you. (1Timothy 6:20-21)

SAYING THAT THE RESURRECTION HAS ALREADY TAKEN PLACE: legontes (PAPMPN) ten anastasin êdê gegonenai (RAN): (1Cor 15:12; Col 3:1)

Saying (3004) (lego) means to speak or talk, with an apparent focus upon content of what is said. Present tense indicates that this wasn't a single event but something these men were continually saying.

Resurrection (386) (anastasis from anístemi = stand up <> ana = up, again + histemi = stand) (Click in depth word study of anastasis) means to come back to life after having once died and refers most naturally to bodily rising from dead. It is interesting that this is the only use of anastasis in Paul's pastoral letters.

Already (2235) (ede) means now, even now, and in reference to time marks an action as completed.

Taken place (1096) (ginomai) caused to come into being or into existence. The means it had happened at a point in time and the effect or results persisted. It speaks of permanence of this occurrence (with the implication that it won't happen in the future!)

If the resurrection had already taken place, it would give the person who heard and believed this false teaching about the resurrection no hope . It was if they were saying "Forget about the rapture. Paul only meant it symbolically and it was not to be taken literally!".

But in fact Paul had assured the Thessalonian saints  they could maintain a steadfast hope because of a literal bodily resurrection and subsequent rapture writing...

13 But we do not want you to be uninformed, brethren, about those who are asleep, that you may not grieve, as do the rest who have no hope.
14 For if we believe that Jesus died and (literal bodily) rose again, even so God will bring with Him those who have fallen asleep in Jesus.
15 For this we say to you by the word of the Lord, that we who are alive, and remain until the coming of the Lord, shall not precede those who have fallen asleep.
16 For the Lord Himself will descend from heaven with a shout, with the voice of the archangel, and with the trumpet of God; and the dead in Christ shall rise first (literal bodily resurrection)
17 Then we who are alive and remain shall be caught up together with them in the clouds to meet the Lord in the air, and thus we shall always be with the Lord.
18 Therefore comfort one another with these words. (See notes 1Thessalonians 4:13; 14; 15; 16; 17; 18)

The literal bodily resurrection is at the crux of Christian faith and so Paul had just instructed (present imperative = command) Timothy to be sure to continually...

Remember Jesus Christ risen from the dead (see note 2 Timothy 2:8)

Jesus Himself testified that His resurrection was a bodily and not a spiritual resurrection...

33 And they (Cleopas and a second person on the road to Emmaus with the resurrected Jesus) arose that very hour and returned to Jerusalem, and found gathered together the eleven and those who were with them,
34 saying, "The Lord has really risen, and has appeared to Simon."
35 And they began to relate their experiences on the road and how He was recognized by them in the breaking of the bread.
36 And while they were telling these things, He Himself stood in their midst.
37 But they were startled and frightened and thought that they were seeing a spirit.
38 And He said to them, "Why are you troubled, and why do doubts arise in your hearts?
39 "See My hands and My feet, that it is I Myself; touch Me and see, for a spirit does not have flesh and bones as you see that I have."
40 <And when He had said this, He showed them His hands and His feet.>
41 And while they still could not believe it for joy and were marveling, He said to them, "Have you anything here to eat?"
42 And they gave Him a piece of a broiled fish;
43 and He took it and ate it before them. (Luke 24:33-43)

About 10 years prior to this second and last epistle to Timothy Paul had written to the Corinthian church explaining that...

12 if Christ is preached, that He has been raised (literally, bodily) from the dead, how do some among you say that there is no resurrection of the dead?
13 But if there is no (literal bodily) resurrection of the dead, not even Christ has been raised;
14 and if Christ has not been raised, then our preaching is vain, your faith also is vain (empty, fruitless = There is no true Christian faith without the literal, bodily resurrection, and thus no hope of forgiveness, salvation, or eternal life).
15 Moreover we are even found to be false witnesses of God, because we witnessed against God that He raised Christ, Whom He did not raise, if in fact the dead are not raised (we apostles would all be lying about God for we have said that God raised Christ from the grave, but that can't be true if there is no resurrection of the dead)
16 For if the dead are not raised, not even Christ has been raised;
17 and if Christ has not been raised, (literally, bodily) your faith is worthless; you are still in your sins.
18 Then those also who have fallen asleep in Christ have perished (all who have died believing in Christ have perished)
19 If we have hoped in Christ in this life only, we are of all men most to be pitied.
20 But now Christ has been raised from the dead, the first fruits of those who are asleep (the first of a great harvest of those who will be raised to life again.) (1Cor 15:12-20)

Hiebert explains that Hymenaeus and Philetus...

denied a future bodily resurrection and taught that the only resurrection there was had already occurred in the spiritual renewal of the believer in regeneration. They probably allegorized ("spiritualized") and thus misapplied Paul's teaching about the believer's spiritual union with Christ in death and resurrection

Romans 6:3 Or do you not know that all of us who have been baptized into Christ Jesus have been baptized into His death? 4 Therefore we have been buried with Him through baptism into death, in order that as Christ was raised from the dead through the glory of the Father, so we too might walk in newness of life. 5 For if we have become united with Him in the likeness of His death, certainly we shall be also in the likeness of His resurrection (See notes on Ro 6:1-3 , 6:4-5)

 and insisted that the doctrine of the resurrection had only a spiritual meaning and application.

They allegorized (Ed note: An allegory is a  form of literature in which a story points to a hidden or symbolic parallel meaning) away the doctrine, and turned all into figure and metaphor (Waterland, quoted in Ellicott).

Accepting the current philosophy that matter is evil, they argued that a physical resurrection was unthinkable. (2 Timothy by D. Edmond Hiebert) (Bolding added)

Nelson Study Bible comments that

This was probably an early form of Gnosticism that emphasized a spiritual resurrection over against the Christian belief in a future bodily resurrection. (Radmacher, E. D., Allen, R. B., & House, H. W. The Nelson Study Bible: NKJV.  Nashville: Thomas Nelson)

KJV Bible Commentary agrees writing that...

Resurrection is past. They probably spiritualized the resurrection of the future as the Gnostics of the day taught. (Dobson, E G, Charles Feinberg, E Hindson, Woodrow Kroll, H L. Wilmington: KJV Bible Commentary: Nelson or Logos)

JFB writes that...

The beginnings of the subsequent Gnostic heresy already existed. They “wrested” (see note 2 Peter 3:16) Paul’s own words (see notes Romans 6:4; Ephesians 2:6; Colossians 2:12) “to their own destruction,” as though the resurrection was merely the spiritual raising of souls from the death of sin. Compare 1Co 15:12, where he shows all our hopes of future glory rest on the literal reality of the resurrection. To believe it past (as the Seleucians or Hermians did, according to Augustine [Epistles, 119.55, To Januarius, 4]), is to deny it in its true sense. (Jamieson, R., Fausset, A. R., Fausset, A. R., Brown, D., & Brown, D. A Commentary, Critical and Explanatory)

Hendriksen comments that Hymenaeus and Philetus...

resembled those present-day liberals who, while refusing to be caught saying, “There is no resurrection,” allegorize (Ed note: An allegory is a  form of literature in which a story points to a hidden or symbolic parallel meaning) the concept. Now it must be admitted that Paul, too, believed in a spiritual resurrection, the act of God whereby he imparts the new life to those who are dead in sins and trespasses (see notes Romans 6:3; 6:4; Ephesians 2:6; Philippians 3:11; Colossians 2:12; Colossians 3:1; and cf. Luke 15:24). But the apostle also most definitely taught the resurrection of the body (1Cor. 15; see note Philippians 3:21), just as Jesus had done (John 5:28). According to Paul’s teaching, denial of the bodily resurrection implies the complete overthrow of faith, for “if there is no resurrection of the dead, then Christ has not been raised either; and if Christ has not been raised, then our preaching is in vain, your faith is in vain, … and you are still in your sins” (1Co 15:13, 14, 17)." (Hendriksen, W., & Kistemaker, S. J. New Testament Commentary Set, 12 Volumes. Grand Rapids: Baker Book House) (Bolding added)

Pastor Steven Cole has an interesting sermon on this section noting that...

it is possible to use the Bible to make progress in ungodliness (2:14, 16-18).

Note the words Paul piles up to drive home this frightening point: “useless,” “ruin of the hearers” (2:14); “further ungodliness” (2:16); “spread like gangrene” (2:17); “gone astray from the truth,” “upset the faith of some” (2:18). The improper use of the Bible is not an innocent, harmless activity. It leaves a trail of carnage of ruined lives in its wake. That’s one reason James 3:1 warns, “Let not many of you become teachers, my brethren, knowing that as such we shall incur a stricter judgment.” That’s why Paul here warns Timothy to “solemnly charge them in the presence of God” (2:14). The Bible is no harmless instrument. It’s a sharp sword and must be handled with proper care....

To use the Bible to teach half-truths as truth is to use it improperly (2:18). These men were not totally wrong. They were teaching a half-truth as if it were the whole truth, which is often Satan’s method. They were teaching that the resurrection already had taken place. They had verses from Paul to back up their views. He wrote often of the fact that Christ is risen  and that we are risen with Him. But he also taught that there is a future resurrection of the body, which these men denied. They argued that the resurrection was only spiritual and thus was an accomplished fact.

You  may wonder, “What’s the big deal? Why is this even worth contending about?” Paul answers that question in 1 Corinthians 15 where he argues that if there is no future, literal, bodily resurrection, then Christ Himself is not even raised and our faith is worthless

Mark it well: Heresy always begins as truth out of balance! There is always an element of truth in the teachings of the cults. That’s how they get their hooks in people. They even have verses to back up their errors. So they prey on the untaught who are looking for “something more” in their faith. But they lead people away from dependence on the living God. If somebody handed you a three-dollar bill with a picture of Frank Sinatra on it, you wouldn’t be fooled. A counterfeit always looks genuine at first glance. That’s why we have to examine the popular worldly teachings cleverly cloaked with the Bible that are flooding the church in our day. They promote half-truths as if they were the truth of God. (2 Timothy 2:14-19 How To Use The Bible)

AND THUS THEY UPSET THE FAITH OF SOME: kai anatrepousin (3PPAI) ten tinon pistin: (14; Mt 15:13; Lu 8:13; 22:31,32; Acts 5:39; 1Cor 11:19; 1Jn 2:19)

Upset (396) (anatrepo from ana = again + trepho = turn) (Click word study of anatrepo) means to cause something to be completely overturned. Paul uses the verb figuratively here of the overturning, subverting or overthrowing of the faith of some by corrupting such vital truths (see notes Romans 6:5, Colossians 3:1; 3:2; 3:3) in addition to denying the encouraging truth of God's promise of the future resurrection when His Son returns for His Bride, the Church (1Th 4:16,17, 18).

Anatrepo is in the present tense indicating that these false teachers are continually overthrowing the faith of certain ones.

In his letter to Titus Paul used this same verb describing

"rebellious men, empty talkers and deceivers, especially those of the circumcision who must be silenced because they are upsetting (anatrepo) whole families, teaching things they should not teach, for the sake of sordid gain." (see notes Titus 1:10; 1:11)

Faith (4102) (pistis) (Click word study on pistis) means a firm persuasion, conviction or belief in the truth.

Genuine saving faith is not just mental assent but a firm conviction, surrender to that truth and conduct emanating from that surrender. In sum, faith shows itself genuine by a changed life.

Wayne Grudem defines faith that saves one's soul...

William Barclay (see critique) defines "faith" as follows

Faith begins with receptivity. It begins when a man is at least willing to listen to the message of the truth. It goes on to mental assent. A man first hears and then agrees that this is true. But mental assent need not issue in action. Many a man knows very well that something is true, but does not change his actions to meet that knowledge. The final stage is when this mental assent becomes total surrender. In full-fledged faith, a man hears the Christian message, agrees that it is true, and then casts himself upon it in a life of total yieldedness. (Barclay, W: The Daily Study Bible Series, Rev. ed. Philadelphia: The Westminster Press)

Faith means relying on what God has done rather than on one’s own efforts. In the Old Testament, faith is rarely mentioned. The word trust is used frequently, and verbs like believe and rely are used to express the right attitude to God. The classic example is Abraham, whose faith was reckoned as righteousness (Ge15:6).

J. B. Lightfoot discusses the concept of faith in his commentary on Galatians. He notes that in Hebrew, Greek, and Latin, the definition of the word for faith