Yeah the rents in some cities seem crazy high. I’m paying 580€/month warm for a 1 bedroom apartment that I live in with my wife so even from minimum wage that would still be affordable but in some places that would probably not be possible.
Working Full time for the new minimum wage you earn about 2400 gross. After taxes and health insurance that is about 1883€. As a single you can very easily live quite comfortably from that unless you are living in a very expensive city.
But couldn’t you just skip the room if that happens? Eventually somebody will be willing to change rooms.
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.