And when the big rock comes or some crazy starts a full scale nuclear war then what? The human race will be trapped and likely extinct because short sighted people were so worried about money that race survival was ignored.
Money is not the issue. Survival and hope are. Humans are explorers not TV watchers. You’ve been watching/reading too much low end scifi.
Those ships are also large enough that the walls can be very thick, preferably packed with soil for growing plants, in order to minimize exposure to cosmic rays.
Such a ship would most likely have to be self-sustaining, i.e. all solids and liquids (and I mean ALL) being recycled and reused indefinitely for very long journeys.
Once you have such a ship, you don't need it to go anywhere in particular. It can just hang about in space for eons.
It would be best, however, if it had a place to go so that some generations hence the remaining humans would be able to live on another planet rather than trapped in the cylinder.
If we spend most of our money now on probes, then we are more likely to find such a place for the cylinder to head towards.
I also wonder if there is anything that could completely destroy a group of people in a sufficiently hardened and outfitted underground bunker. With the advent of VR technology, we could have about 100k people dispersed into various bunkers around the world with sufficient food, water, energy, and entertainment to keep them physically and psychically happy for decades to come.
And if we’re talking Sci-Fi, then it might already be the case that you are in a bunker having survived a massive surface-life-destroying meteor shower and just don’t realize it.