You are probably overstating the dangers of nuclear warming a bit (but not that much). But there is certainly nothing we can't adapt to if we have to. And of course "civilization as we know it" isn't going to be around 20 years from now no matter what happens: 20 years ago, we didn't have the Internet, Cell Phones, GPS and it just took me 20 seconds to think of that. My mother's house didn't have indoor plumbing when she was a girl!
But the important point is that you have no idea how dangerous nuclear waste really is.
Nuclear waste remains dangerous for more than million years. The "design" of nuclear waste disposal facilities involves making an educated guess as to how long it will be before the containers break down and then guessing whether the waste will decay enough before it gets into the water supply. So we are talking about the water supply for a huge geographical area being poisoned essentially forever, not a few people ignoring signs.
Speaking of signs, how do you write a sign an average person will be able to read 100,000 years from now (or even 1000 years from now)? You would need to predict 100,000 years of language evolution! The reality is that even within 1000 years, the only people who can read what we now call English will be scholars!
This just demonstrates how naive it is to assume we can construct anything that lasts for a million years.
In case you are curious, the Yucca Mountain plan was clever enough to realize they couldn't label the site as dangerous; so, they decided it was good enough if a typical drill bit wasn't likely to penetrate the containers when somebody drilled into the waste disposal site -- Mind you these are the same containers they know will deteriorate within one or two half-lives of the stuff inside.