You don’t need FTL travel or multi-generational ships really. If you get close enough to the speed of light, time dilation can make the travel time for the people on the ship be as short as you want. Only for the people that stay on earth would the spaceship appear to take generations. Getting enough energy to reach such high speeds would be the difficult part as well as accelerating quickly without crushing the travellers.

But couldn’t you just skip the room if that happens? Eventually somebody will be willing to change rooms.

Closing the lid doesn’t reduce the number of particles that get into the air.