We could probably build something that could transport a lot of people, at immense cost of course. But it would probably not be a generation ship, because the ship its ecology or its energy source would break down far before they get anywhere. So I guess that's a
No by me.
I worry that the only actually feasible generation ship might look somewhat like this:
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Or worse, like this
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The big challenge with space travel now already is how to put enough fuel aboard to get anywhere. But it gets worse when you talk generations, because you also need to ensure a steady supply of air, food, water and gravity, and an energy source that works for centuries. All the devices you need to keep this going will need replacements and spare parts. If anything breaks and you don't have what you need to repair it, it will at some point need centuries or maybe even millenia before you can even start looking for it.
How many devices do you know that you would trust to keep working for decades before they break down and need repairs? Probably none, or only simple mechanical devices.
Imagine going for one year with your car through the US. But you never go for a pit stop. You can't visit any gas stations or garages. You can't ask anyone for help. You have to fix everything that comes up yourself. If a tire is punctured? You got to have the replacement with you. If you need to fill up the tank, you have to have the fuel with you. If you want to eat something, you need to carry the food with you. Imagine what kind of "car" you would build to accomplish that.
And now imagine you want to build a convoy of 1,000 cars or a bus to carry a 1,000 people, or 10,000 people.
And now you need to scale that up for stuff like lack of air, heat/sun or gravity.
I think one thing is fairly certain though - if we ever build a generation ship, we don't need existing habitable planets anymore. We might still prefer to have them, but we can probably do without them.