What about the passage which says that ‘things which are seen are temporary but that which is unseen is eternal’? Does this use the word ‘Aionios’? It would almost certainly mean unending in this context 
The ref is to 2 Cor 4:16-18ff.
In this case the context would include ‘unending’ (probably), yes. It’s hardly anything to be sad about, though. 
“For the momentary lightness of our affliction is producing for us a transcendently transcendent eonian burden of glory, at our noting, not what is being observed, but what is not being observed; for what is being observed is temporary, yet what is not being observed is eonian.”
St. Paul continues (starting what we call chp 5): “For we are aware that, if our terrestrial tabernacle-house should be demolished, we have a building out of God–a house, not made by hands, eonian in the heavens. For in this also we are groaning, longing to be dressed in our habitation which is out of heaven.”
The use of “eonian” here still corresponds well enough to the concept of that-which-is-being-described being ‘from God’. It may seem redundant to say so in this passage; but Paul is undeniably laying on the redundant repetitions and rephrasing pretty thickly round there in other regards–probably for emphasizing how important and great this all is.
I agree with Jason’s take, but there is also another alternative to “everlasting”. I’m not saying I agree, but it may mean “age pertaining”. I don’t mean that the unseen things will merely “last an age”, but instead the unseen things pertain to an age to come.
Before getting to that, I want to repeat that I agree with Jason that aionios probably meant something like “from God”, “godly”, or “having its causal source in the eternal God” (that last one is Talbott’s).
Anyway, Seth Tipton writes:
See mercifultruth.com/eternity-detailedstudy.htm
Paul may have been saying something like, “okay, the pain is seen, but the glory that you do not yet see - the unseen things - will appear in the age to come”. Again, I’m not saying I agree with that. But it’s possible.
I’m sure St. Paul would have strongly agreed with that, too, btw.
this is an excerpt of an essay I once wrote, maybe it helps anybody:
The word proskairos also occurs in Matthew 13:21, Mark 4:17, and Hebrews 11:25
I’ll show these verses now in several translations, before I turn to 2 Corinthians 4:18 in detail:
Matthew 13:21
“But since he has no root, he lasts only a short time (Gr. proskairos). When trouble or persecution comes because of the word, he quickly falls away.” (New International Version)
“Yet hath he not root in himself, but dureth for a while: for when tribulation or persecution
ariseth because of the word, by and by he is offended.” (King James Version)
Mark 4:17
“But they have no roots. So they last only a short time(Gr. proskairos). They quickly fall away from the faith when trouble or suffering comes because of the message.” (NIRV)
Hebrews 11:25
“Choosing rather to suffer affliction with the people of God, than to enjoy the pleasures of sin
for **a season **(Gr. proskairos);…” (King James Version)
“He chose to be ill-treated along with the people of God rather than to enjoy the pleasures of
sin for a short time.” (New International Version)
In all these occurrences proskairos rather seems to mean a short time, a season or a while.
An online dictionary, it seems to be the Liddell Scott; gives the following meaning for proskairos: for a season, temporary
“season” is defined in an English dictionary: A period of time not very long; a while; a time.
Before I turn to 2 Corinthians 4:18, remember Mr. Darby wrote, “But this does not alter the meaning of the word: aionios is properly the opposite to proskairos.”
But what is the opposite of “a period of time not very long”? Of course eternity would be the opposite of such a period, but also one or several long ages, or even a single century would be the full contrast to a short season, while the contrast of eternity is time and not a season.
The verse of interest is:
2 Corinthians 4:18
mê skopountõn êmõn ta blepomena alla ta mê blepomena ta gar blepomena proskaira ta de mê blepomena aiõnia
“So we don’t spend all our time looking at what we can see. Instead, we look at what we can’t see. What can be seen lasts only a short time. But what can’t be seen will last forever.” (NIRV)
As I said proskairos is not time itself, this would be chronos, but rather a (shorter) period as already shown, you can contrast strict opposites or things that are related to each other, you can contrast a lake with a desert and you can contrast a lake with an ocean; one might suppose here is contrasted time with timeless eternity in a Platonic sense, but you can also contrast something which lasts only a short present period with something that is yet future and will last for ages, as I already said.
If Paul would have contrasted time with eternity I think he would have written,
“For the things which are seen are ‘chronikos’ (temporary/temporal, in the sense of pertaining to time);
but the things which are not seen are eternal (pertaining to eternity).”
But Paul did not use chronikos, the adjective of chronos, time, which I think he would have done, if he had intended to contrast time with eternity here, but he used proskairos, which is not related to time (chronos) itself, but rather means a season as shown. I think this verse proofs not neccessarily that aionios should be understood as infinite duration, because it does simply not explicitly say so. It might be the perfect definition of Hebrew olam which means something like “hidden time” as far as I know:
Things that are seen last only for a (short present) period (Gr. proskairos),
but things (yet future), not seen (yet and with an unavowed end), are lasting for [a] (long future) age[s] (Gr. aionios).
This might be a possible interpretation without any relation to timeless eternity or endlessness.
I agree with Sven. Proskairos is not being used by Paul to contrast limited time with unlimited time. It is simply used to contrast a shorter measure of time (the passing of which is observable) with a longer measure of time (the passing of which is not observable) - i.e., the soon-to-commence Messianic age, which has already lasted nearly 2,000 years. Paul is contrasting the present period of hardship and suffering he and other believers were experiencing, with the happiness they would enjoy at the resurrection, at the end of the Messianic age. He represents their resurrection bodies as being already in existence (kept “in the heavens” - 5:1) for an aionion duration of time because it is so certain that we are to be “clothed” with them - and since they are incorruptible, Paul pictures them as enduring throughout the Messianic age until the time comes for mortals to finally acquire possession of them.