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We have a world in the first age (think Atlantis without tech but with magic, a civilization with wealth, overflow of resources, no sickness, no pain, a civilization with more power than everyone else, enough food for everyone, never a bad harvest, never a parent or child lost to child birth, weapons forged by the gods themselves. Warriors return home from battle since the armor is invincible.), gods walks the earth, assist the humanoids in the world with different tasks in return for sacrifices and faith (the life force of the gods).

Gods and humanoids live in synergy, (Gods helps the humans thrive, humans believe in gods, gods get power from faith, power is used to help humans, full circle)

There are no evil gods nor good gods; gods are filling a purpose. God of life, grants good births and healthy children. God of death carries brave people over to the other side safe. God of harvest helps in bad years etc. God of war gives legendary weapons and armor to people. And the weave of the gods provides the magic in the world.

Now let's say that these gods someday just disappear, in an instant, something (to be decided) happens in the plane of the gods so they all disappear (since they technically cannot die). This means they no longer walk among men, no longer provide good births and strong children, no more good harvest in bad years, and dead people are no longer guaranteed safe passage to the other side. But most of all, all magic is gone, no more healing spells, no more fast travels no more than "what we would have" in a fantasy setting.

This is resulting in crops that were sustained by the gods no longer growing to the extent they might have (good crops still grow good), but bad crops dry out, get pests, fungus. Weapons are just steel, no longer imbued by the power of the gods, armor is just thin plate sheets (swords are still sharp and armor still protects, but no better than ordinary armor). People wounded are getting ill from infections an can no longer be instantly healed.

Potential other effects:
Priests and Clerics are losing their offices, (might even be called frauds). People start fighting for food. Warriors who were invincible start questioning if it is worth the risk. Cities start getting overrun by wild life and other fantasy creatures that didn't want to attack before. And with reduced power of the military both in strength and numbers, that could be a problem.

Q: Would this event be enough to create a post-apocalyptic event? (Analog to fallout-fantasy)

(For reference, the story is in the third age, where the first is the golden age, the second is rebuilding, and the third is the current, haven't gotten any cool names for the ages yet.)

Humans will not survive.

Each of the analogues presented by previous answer is, IMO, wrong. They keys thing is, humans lack knowledge. Bad crops will die you say, but good crops will survive. But humans don't know the difference between good crops and bad crops, because all crops were good before. Noone has any experience with seeding seeds from "bad crops". Noone has ever used irrigation to water their crops. They not only lack the knowledge of how to build an irrigation system -- they even lack the knowledge that there is a benefit for them. Crops would always grow everywhere, regardless on how good or bad the soil was, and noone has ever used manure on their fields; there doesn't exist any knowledge that this has a benefit. Same for health care. Even people never were sick, noone has even an inkling of a clue on how to deal with it.

People will be worse off than in any zombie invasion you can imagine. And that's due to the lack of knowledge on how to exist.

While the collapse of the preexisting social order is quite plausible, total extinction is pretty unlikely. Here's a likely end-of-civilisation-inducing scenario. It won't do anything drastic like wipe all life off the face of the earth, but it'll probably do pretty well in dragging humans into a dark age:

Wannabe Gods

The absence of the gods creates what we'd call a market imbalance. Once it's noticed, sooner or later the thought will enter human heads: why not become the gods? Someone has to fill the gap in service coverage, right?

Priests will come up with all kinds of explanations to justify all kinds of convoluted rituals, but in the end it'll just boil down to finger pointing: they(worshippers of another god) dunnit. Warriors would become tyrants who rule over the common folk by force. No matter their background, they'll all claim to "make the world godly again" but fail to actually do so. In true medieval fashion, supposed rituals of ascension would probably involve copious amounts of depravity/carnage, like bathing in the blood of a thousand innocents killed.

Cue the fragmentation and break down of society as warbands led by self-proclaimed Prophets and God-Emperors and such carve the land up and declare holy war on each other. The line between clergy and warlords would probably cease to exist here. The madness will last until people truly accept that the gods are gone for good, but that can take quite a while as concrete evidence of their (past) existence and influence is plentiful

Unlikely (but not impossible)

You could just make a parallel with both 1929 and 2008 crises. The market was screwed up and hell broke loose because of that. Of course the scale of these events is nothing compared to not being able to produce food or have healthy children, but they were, in a way, cataclysmic events in their own proportions.

But the point is that people recovered from that.

The world didn't end just because its financial bedrock was crushed. It was bad, people lost their homes, unemployment skyrocketed, suicides, etc. It took time but eventually things got back to some degree of normality in the years that followed.

Humans are hard to beat. Unless gods and magic are something humans can't live without (meaning no medicine to replace healing spells, no horses and carriages to replace teleportation, no agriculture techniques to guarantee food production, etc), they should be fine.

There are two aspects in this answer, a short-term one and a long-term one.

At short term, right after the disappearance of the gods, you would expect a lot of faithful followers of these gods to pray for their return, as praying and sacrificing as been the best way to proceed in case of emergency and crisis ever since people remember and tell tales. Gradually, more and more would lose faith in the gods' return, as time passes and crises amass: more death in child birth or children dying at young age, crop harvest failing maybe not in the first, but in one of the following years resulting in starvation, emigration, uprisals and civil war. You cannot support as large cities and militaries as before. People will emigrate to the country side, trying to scavenge food or steal from the farmers surrounding the cities who will have to defend themselves and their remaining crops.

You can take the decline of the roman empire and its fall under the onset of gothic tribes desparately migrating westwards driven by nomadic tribes conquering their lands as a historic example. This has basically set back the economic and cultural development on the European continent by centuries. Although it was not an apocalyse, it possibly appeared so to the people living at that time, constantly starving and running for their lives.

How well your society can recover, will depend on external threats (as marauding trobes in the case of Ancient Rome) as well as your people's willingness to help out one another. If people are rather willing to suffer personal disadvantages for the sake of 'the greater good', as the survival of their people, e.g. paying high taxes to support the poor in society, offering public hygiene and basic medical service to everyone, public schooling etc., every young man or even woman serving in the military or the public services for some time in their lives as a service to society, then you can buffer the impacts much better than if people fight only for themselves, abandoning all mercy, compassion and support for the weaker ones when faced with hunger, civil unrest and the loss of their beloved gods.

The answer to this depends on details that are not in the question. This really depends 100% on what the humans do while they are in synergy with the gods, and how the gods actually help them.

Try this as an experiment. Get a friend and find a door. Have your friend on the outside of the door and put yourself on the inside. Now close the door and kind of lean against it, so that you're pushing on the door in the opening direction, but it's held closed by the latch. Use the door to prop yourself up a bit. You'll be here a while. When you least expect it, the friend opens the door suddenly.

Did you fall? Or did you just stumble?

Now let's repeat this experiment. Only now, lean a bit harder into the door, and we're going to do a task. Repeatedly take your right shoe off and put it back on. This is a hard task to do while standing. Not everyone can do it without losing balance. But it's pretty easy to do while leaning against the door.

Now what happens to you when your friend opens the door? I highly recommend having some padding on the other side so that falling doesn't hurt.

So the real question is whether your society chooses to just lean on the gods, or if your society relies on them to stay standing. This is very society dependent. No two societies will really answer the same way.

Consider the USA. I'd say we lean pretty heavily on the internet. If it were to disappear, we'd stumble pretty hard. It'd be bad news. But we'd probably roll with it. We'd probably catch ourselves.

Now consider what would happen if our power disappeared. All fossil fuels and all existing renewable electricity devices vanish. I think we're going to do a lot worse than stumble. Our entire agricultural system depends on electricity and diesel fuel. There will be mass starvation, and that tends to earn the "apocalypse" title rather quickly in most people's vernacular.

Do your gods bring to the table something like the internet? Or is it being treated more like electricity and fossil fuels?

It would be rough for the humans. It is described in the Book of Genesis.

Eden was nice. Delicious plants grew on their own. You did not have to do much. No clothes either!

http://www.vatican.va/archive/bible/genesis/documents/bible_genesis_en.html

[2:8] And the LORD God planted a garden in Eden, in the east; and
there he put the man whom he had formed. [2:9] Out of the ground the
LORD God made to grow every tree that is pleasant to the sight and
good for food, the tree of life also in the midst of the garden, and
the tree of the knowledge of good and evil.

But God got mad when the humans ate of the tree of the knowledge of good and evil. God did not leave, but he sent the humans away from His presence and from beautiful Eden He had maintained to a wasteland without divine maintenance. Humans would have to work. And they would have to suffer. Also they would have to deal with weeds.

[3:17] And to the man he said, "Because you have listened to the voice
of your wife, and have eaten of the tree about which I commanded you,
'You shall not eat of it,' cursed is the ground because of you; in
toil you shall eat of it all the days of your life; [3:18] thorns and
thistles it shall bring forth for you; and you shall eat the plants of
the field. [3:19] By the sweat of your face you shall eat bread until
you return to the ground, for out of it you were taken; you are dust,
and to dust you shall return."

Side note - I am interested to know who the LORD God is talking to in this passage.

[3:22] Then the LORD God said, "See, the man has become like one of
us, knowing good and evil; and now, he might reach out his hand and
take also from the tree of life, and eat, and live forever"--

In any case, so too your scenario of the gods departing. The gods leave and no longer maintain things and make life easy - an apocalypse in a sense. The humans will have to work harder, and suffer, and deal with weeds. Maybe it will make them stronger and craftier in the long run. It would be very hard in the short run.