If the ship requires human crews to conduct regular repairs, then the ship probably won't last long enough to get where it is going. A ship that is going to last for 10,000 years will need to be built like an Egyptian pyramid, solid enough to last forever and be functional despite millennia of erosion. Putting someone on that pyramid to conduct repairs might sound good, but after 10,000 years those repairs would become just another source of erosion. The activities of that one person would wear down any system with which they interact.

http://www.projectrho.com/public_html/rocket/enginelist3.php

But even if you ignore that limit, at that kind of speed you also need to account for your outer hull undergoing significant damage as each stray intergalactic hydrogen you hit has a kinetic energy of 34.75 TeV in your frame of reference, and while this will create a large supply of antimatter, it will do so in a way significantly less helpful to your goal than the fact we can transmute lead into gold by using the former as a radiation shield in a nuclear reactor is useful to someone who wants to get rich quick.

Bonus fact: this energy level is higher than the LHC, so you may have some extra weird mass-modification effects besides relativity as the front of your spaceship is now a Higgs Boson torch.

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

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?

(1) we don't really know the threshold yet to be beyond repair I think. It may be 1% but may be 20%.

(2) as others noted, it's 15 years for 0.1% damage using your math.

(3) your timeline directly contradicts the article doesn't it, which suggests keeping people in ideal conditions would have a DNA half life of millions of years.

(4) it's not a given that you need to wake people up to repair DNA. Maybe you just need to bring them to 1 degree Celsius or something. Or maybe nanobots can work on you while completely frozen. I think we are quite far from knowing right now.

So yes, for long term travel DNA repair will be important but how it actually works I think we know little about.

Then you only need a crew of X people at any given time to keep the ship operational. Some generations would be born and die during the trip (kinda depressing, but I like to think I would be happy in that scenario)

If you don't like the idea of having a machine as your mother, you could also do IVF with the women that are alive and well and constitute ~50% of the crew keeping the ship operational

  >>> log(0.99, (1/2)**(1/10000))
  144.99569695122509

[1] It may be a bit worse in the long run, since if many bonds break, the higher order DNA structure gets damaged and that may change the rate of decay experienced by individual bonds (?).

If indeed 1% break after 15 years, then 10% (ignoring compounding) break after 150 years, and 50% after 750 years. That doesn’t square with a half life of 10,000.

https://www.researchgate.net/post/How-long-can-fresh-frozen-...