Well it's unlikely we will ever get even remotely close to the speed of light we won't make it several million years in space either.
Space is a death trap, if radiation doesn't get you even just a little it is likely to end up sterilizing the first gen space travellers, then you will need artificial gravity it is a must, food, water, oxygen if any of the systems fail you are screwed. The size of the ship will likely need to be the size of earth it self to just accommodate all the system to keep people alive there isn't enough resources on earth to do that and it will need to be some thing that is build in orbit, you will need to start mining the outer solar system.
propulsion is another problem, even if you can get up to the speed of light it will still take millions of years, the human body nor the ship it self will withstand those forces even in zero gravity, however considering the size of the ship it will have its own gravitational field.
No matter how much technology and understanding of the universe improves the science will never be able to support it not in our life time not in a million life times.
The physical problems with interstellar travel is just too immense, just making it out of the solar system will be a feat in it self and we only recently managed to get to the outer solar system and not even completely out of it even and that took nearly 70 years.
While not impossible just improbable and will take several million generations to do it so you will need to deal with either negative or positive population growth not forgetting the several 100 thousand even millions needed to keep a ship like that in operation, even with automation things are going to break down and not be fixable.
just one stupid small mistake and your trip is over, you are just better off slapping rockets on the earth move people deep underground and launch the entire earth in the direction you want to go. lol It is fun to think off but the reality is much more sobering once you consider just how astronomical the problems are.
not pessimistic just a realist lol