Just imagine our technological capability in 100 years, much less 1000.
However, I think it's likely we go thru another dark ages at some point. Although with computers and artificial intelligence I'm not really sure how much we would actually lose. But some event is going to wipe out 3/4 of the population - be it a global pandemic, wars over limited resources, an asteroid or super volcano blotting out the sun and causing global famine.
I think in a few hundred years we could see a generational starship - built entirely in outerspace - setting out with 50,000 people to go colonize a planet. Long ways to go, but if we get to something like 10% the speed of light we could conceivably reach a habitable planet after a few centuries. The biggest challenge would probably be that you have to be able to fabricate and refurbish basically every part of the ship. Although I doubt the power requirements to achieve even 10% of the speed of light are remotely practical.
We will probably figure out a way to terraform Mars before that, though (we probably think we already know how to do this). That would at least protect us from an extinction event.