I need an in-character incentive

[Gnaritas]

For an upcoming campaign i want to create a world similar to Dark Sun, but even less civilization. More like Fallout.

But i wonder how to motivate my players into adventuring. It's not that i doubt that they will start adventuring, but opposed to in a "normal" D&D-world, i can imagine players wanting to stay in a safe haven should they find it.
I need a reason for them to keep going.

So far i have considered 3 options to start the game.
1. They are some sort of walled town or castle with a bunch of other residents, probably for generations. They are self-sufficient and not even scouting parties go outside, they do have to fight off attackers and very rarely a tradesperson comes by.
2. Same as above, except that they live in a large cave and most of the residents, including the party, know that there is a world outside of the cave. (The players will obviously suspect it)
3. I dump the players into the world, make a backstory how you ended up together, now "survive".

I don't like option 3 much, and option 2 perhibits me to give full details of the world and history at the start of the game. So that leaves number 1, or any other suggestion you people can give me.

As a reason to go out, i thought i would let the water supply drop and they have to figure it out. As they do that, the castle is overrun when they get back and their families have either been slaughtered or captured by Sand Pirates.
At least now they have a reason to keep going, but i don't like this story much, seems a bit standard.

Does anyone have a good idea?

[Certified]

Working with option 1, a simple solution is an attack from an unknown source. This might be a rampaging beasty that destroys the wall, or segments of it and while there is a town champion and monster is slain, the players are selected to help bring back rock, and materials to rebuild the wall. Alternatively, the champion is wounded, say poisoned, negative energy, or something requiring a mid to high level cleric. PCs then need to acquire and bring back a healer. To bring the devastation level higher, this might be more like a force of nature that levels the town. The PCs then need to either help survivors or may be the only survivors.

Replacing the beasty with an army works also, and allows for things like the PCs beginning as slaves, and the opening story being their escape or sale into a caravan or arena.

[Gnaritas]

Hmm...those are ideas i could work with, thank you!

[Shining Wrath]

Refugees reach Home from another castle / cave, bearing tales of slaughter. The party is selected to investigate and finds out that Evil Is On The Move. In order to preserve everything they know and everyone they love, they must find a way to stop Evil before Evil reaches Home.

The local cleric / shaman receives a vision, which is also visited upon the members of the party. The cleric realizes that the party members have been called by the Entity Upstairs to go forth and bring back something that Home will need to [stay warm | defend themselves | feed themselves | cure the plague that is sweeping the land | get premium cable].

The population of Home has been slowly increasing. The characters are chosen by lot to be exiled to bring the population back into line with resources.

[Tovec]

I think you need to put more thought into your status quo. Are the walls sealed, letting no one and nothing in or out? Or do travelers come from far away, but stick to existing trading paths? How small/big is the safe-zone. What is the safe-zone and its features.

THEN you need to figure out the inciting incident, has it been a long wearing down of resources over time, or a sudden unexpected intruder from outside?

Sorry in advance if I missed something and this doesn't help.

[Zubrowka74]

The only well in the village/cave/settlement/refuge camp dries up. They have to go in the world and find a new place, perhaps another community or a new location altogether.

[Fable Wright]

The only well in the village/cave/settlement/refuge camp dries up. They have to go in the world and find a new place, perhaps another community or a new location altogether.

This. Easiest way to start the game, and makes logical sense.

Otherwise, having some sort of incurable and lethal disease like Mummy Rot spreading through the town, forcing people to evacuate or die, including the PCs.

If you think having some form of catastrophe separating them from the town is too cliche, you might consider their absence a ritual thing. When there are too many people for the town to support with their resources, they send unmarried men over the age of, say, 20, out into the desert to discover a new place to settle. It makes sense from a practical point of view (though the villagers don't explain it to each other this way), as anyone who has an essential job has some form of respect or job security that would help them get married, and if they're just doing a job that's redundant for the town, best to rid it of the drain on resources, with the potential of random survivors coming back with a cornucopia of resources for future expansion. It also helps as a hook for roleplaying, because they all have to have some reason why they didn't get hitched, usually some form of personality flaw that can be used to lead to character development.

[ArcturusV]

Of course another option I haven't seen yet is that the players get forcibly exiled. It has similar flavoring to option 3 the OP put up, but not quite the same. There is a town, they get to know the town a little, maybe do a single "urban" adventure in the outpost, etc. Something happens where they are told to leave on pain of death. What this Something is can vary. I mean knowing most players I've seen, they'll do SOMETHING during that one "urban" adventure to deserve Exile. It might be a frame job. It might be mistaken identity. It might just be them pissing off someone that the locals don't want to protect you against.

It gives the "survival" flair, but also gives them a potential long term quest hook to clear themselves, burn down the settlement in revenge, whatever.

[Deadline]

So far i have considered 3 options to start the game.
1. They are some sort of walled town or castle with a bunch of other residents, probably for generations. They are self-sufficient and not even scouting parties go outside, they do have to fight off attackers and very rarely a tradesperson comes by.

The only problem with this item is that you are claiming they are self-sufficient. Remove that, and you have a perfect reason to go adventuring: they need to routinely send out groups to gather/trade for supplies. They could even be mostly self-sufficient or, as others have noted, the self-sufficiency suddenly becomes compromised (the well dries up, crops die, etc.).