This is part of my Exegetical Compilation Project, which can be found here.
Hebrews 9:27 is a famous verse commonly quoted against hope of any post-mortem salvation (universalistic or otherwise). “Just as man is appointed (literally “laid up”) to die once, and after that, judgment…” Opponents say that an extra chance after death must be speculated to be something extraneous to this verse, but so is hopelessness of the judgment: the verse itself does not indicate that the “crisis” {krisis} is hopeless. One way or the other there are more details in which light the verse should be understood, including elsewhere in EpistHebrews.
The local preceding context itself is about how previous high priests, even if they kept off judgment for the people by sacrificing something other than themselves, still were mortal and died. By contrast, Christ sacrifices Himself to put the covenant of salvation in effect, since a covenant is never in force while the one who made it lives but is valid only when the one who makes it dies (9:16-17)–which is why those who could not live after dying sacrificed other lives belonging to them in representation of themselves. And yet Christ lives eternally to put that covenant of salvation in effect: a covenant God makes with Israel, which Israel is supposed to keep, but which the Son (acting as the perfect Israel, the perfect prince of God) perfectly keeps and puts into effect.
Thus the contrast by comparison: just as it is appointed for men to die once and after this a crisis (for those men, since they cannot come back to life under their own power), so Christ (verse 28) also having been offered once to bear the sins of many (which in other contexts means “the sins of all”, as any Arminian would agree) shall be seen a second time, apart from sin, by the ones awaiting Him into salvation.
Consequently, the judgment or crisis mentioned by the Hebraist at verse 9:27 is contrasted explicitly to the superior salvation from sin that Christ promises by His covenant, sealed by His dying and rising again: men die once and then are in crisis–a judgment from God (including as the Son) due to our sin–but Christ (the Judge Himself) dies once and lives again to save sinners from our sins! Which is exactly why Christians should eagerly await His second coming when He shall be seen by everyone!–even though that will also result in crisis-judgment for many people.
And what is the covenant that Christ puts into effect by dying and yet living? The Hebraist talks about it at 10:16, quoting Jer 31:33, “This is the covenant that I will make with them, after those days, says YHWH” (referring to the days of Israel’s punishment for her sins and the coming Day of the Lord). “I will put My laws upon their heart, and upon their mind I will write them. And their sins and their rebellions I will remember no more.” “Now where there is forgiveness of these things,” comments the Hebraist, “there is no longer an offering for sin.”
If the Father and the Son do not keep acting in solidarity with that covenant They have made with each other, as a promissory to the covenant YHWH will eventually make with penitent Israel after their days of punishment, then They are breaking covenant with each other, which would put Them on par with sinners who break their covenants with God. A mere static establishment isn’t enough, just like a promise to keep the covenant isn’t enough for a human: They have to perform, and to keep performing. And the Hebraist emphasizes that this covenant which will be made by God with penitent and previously punished Israel in the Day of the Lord to come, was first put into true and perfect effect as a covenant between Son and Father with the death of Christ (the Son being faithful unto death for the Father, and the Father being faithful beyond death for the Son).
To cease seeking, or never to seek, to bring about salvation of sinners from sin, would be for the Persons of God to break covenant with each other on that topic, too.
See also comments on Galatians 3.
This has a lot of connection to why and how the Hebraist continues on into (what we now call) chapter 10 with a discussion of why Christ is superior to high priests; and a big part of the Hebraist’s argument is similar to the previously-mentioned argument made by Paul in some other places (notably in Galatians): Jesus sacrifices Himself as a descendant of Abraham to keep up Abraham’s side of the Abrahamic covenant, for the sake of all descendants of Abraham (which are all rational creatures since God Incarnated as a descendant of Abraham) who have rebelled and thus broken the covenant which Abraham intended to make but which God graciously spared him from actually going through with, meaning the Father and the Son made the covenant between themselves.
That means the covenant (of the promise, unlike the Mosaic covenant) cannot be broken by anyone’s sin (because Abraham didn’t actually participate, so the covenant was only about Abraham and his descendants, between the Father and the Son); it can only be broken by either the Son or the Father, neither of Whom are ever going to break covenant with each other.
But because the Son stood in as a descendant of Abraham, for Abraham and all of Abraham’s descendants (i.e. every created person, numbering as many as the stars in the sky or grains of sand at the sea, poetically speaking), if any person sins then the Son is the one who pays for that sin, requiring the death of the Son despite Him being sinless Himself (because that was the type of covenant made, passing between hewn animals to show that if either party breaks the covenant they’ll be slain like the animals).
But since the Son is sinless Himself, the point of dying wasn’t to satiate the wrath of God, but rather to keep the covenant: the covenant between Father and Son only breaks if the Son refuses to sacrifice Himself for the sins of other people. And the promise given was that all Abraham’s descendants would be led finally into righteousness.
That means the passion on the cross, among other things, is an enacted assurance that God intends to fulfill His promise to Abraham and to Abraham’s descendants (who are all persons created by God, thanks to the Son Incarnating as a descendant of Abraham): the promise being to reconcile all things to God which need reconciliation, whatever those things are, whether things in the heavens (i.e. rebel angels) or things on the earth (rebel humans for example) or things under the earth (currently dead humans and even slain rebel angels). And if we have been reconciled to God through the death of His Son (as Paul says in Romans 5), how much moreso shall we be made alive into His life! In other words, there is no reconciliation that ends with permanent sinners, or with annihilated sinners, or with people no longer sinning but somehow not sharing in God’s own eonian life.
For a more detailed look at the grammar of the verse in question, see my remarks from an early article here: viewtopic.php?f=14&t=277&p=95070#p1931 (until I remember to get around to bringing it over. 
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As always, members are invited to discuss interpretations of these verses below, and to link to discussions either here on the forum or elsewhere.
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