Here's the problem, you have to bend space the opposite direction it does from mass to make it work. For that, you need antigeavity. And the only way to make antigravity, is with negative energy. Which is a real thing that actually exists. Basically, the universe runs on averages. So long as a system averages to a number that works, discrete parts of it can have values that don't make sense, so long as the rest of the system makes enough sense for the average of it to be sensible. So in a system that hovers around 0K, for example, it's possible to have tiny fluctuations that occasionally dip to negative temperatures. The math gets weird, but generally it doesn't matter, because those regions are too tiny and random to make any use of it.

But, theoretically, it is possible to harness negative energy. It's been a while since I looked into it, but IIRC, the best theory is to basically concentrate an enormous, mind boggling, ludicrous amount of energy, and then at the very edges of that system you should be able to bleed off tiny bits of negative energy fairly reliably. But we're talking civilizations that move stars tech here. I think the idea was for a giant ring, that would encompass our solar system, kuiper belt and all, and get it to spin. The amount if energy required to spin something that large is mind boggling, and that's your high energy system, then along the surface you can bleed off negative energy. But even that would be an insanely tiny trickle of negative energy. Unless some new method of bending spacetime is discovered, Alcubierre is just unfeasible. However, this could be more practical for wormholes. But even still, likely looking at a microscopic event horizon for the giant ring, it would be for communication only. But at least you can still technically scale up large scale systems like this to theoretically make something large enough for a person to enter.

That's the problem though. While antimatter exists, which has negative mass, it exists only in small amounts, and you'd have to have a massive amount of it to accomplish such a feat. We'd need to find a way to create it.

And don't get me started on the other problematic aspects of it, like space debris.

I love PBS Space-time