A planet. You're describing a planet. And we're doing a pretty terrible job keeping this planet we're traveling on sustainable for human habitation.
Add a gas giant or two (or 4) to protect it against incoming space debris and help stabilize orbits.
Add some neighboring rocks to also help with that + give them extra material to use once they arrive.
Implant ideas of leaving the rock and coming from a place of great power into the colony’s generational memory.
Include some infrastructure to aid them.
Even the worst outcomes predicted by climate change models leave some inhabitable regions right? Definitely not sustainable for current population levels though...
Anyone got a back-of-the-envelope calculation for what size asteroid we'd need for this?
As long as it can support crops, oxygen generation, etc the passengers just proceed to live a “normal” life for the next couple generations until they arrive in orbit around a brand new star.
The hardest part would be maintaining culture to still identify with the purpose of the mission after there’s no one of the original passengers left over