The US produces less than half the volume of an Olympic-sized swimming pool of nuclear waste per year in total, so it's not exactly hard to manage. Wind and even solar take up a lot more space than nuclear for the same energy, even if we were to consider decades worth of nuclear waste storage. Nuclear power production has about 130x higher density than wind, and needs 34x less space than solar PV.

And that's considering that the US doesn't even use their used nuclear fuel efficiently like, say, France. 96% of French nuclear fuel is recycled by them, while the US doesn't really recycle their nuclear fuel. Thanks to free market capitalism fuel recycling never got commercialized in the US, so the over of century of usable fuel we have in recyclable nuclear fuel is just wasted. It's cheaper to just buy new fuel rather than recycle, so of course companies don't recycle. American problems I guess.

If space were a big issue than nuclear would still win by a long shot even over the long-term. There's very little of it produced, it doesn't take up much space to properly and safely store for tens to hundreds of thousands of years, and the power production is extremely reliable so you don't need miles upon miles of giant batteries to store excess power just in case.