If it were actually a problem, someone would have to do something about it. The volumes are so small they can simply be stored in an ad-hoc matter.
It is a mystery why people get so fixated on the waste. It is very hard to articulate a problem apart from the fact that it goes away eventually (unlike most waste which sits there pretty much forever). We produce a lot of dangerous substances; nuclear waste isn't the worst of them.
Don't forget that there are already nuclear waste sites that are managed by the government. It's just a small operational cost for today's governments.
There doesn't seem to be a believe that governments will be around for 10000 years to keep this managed securely. But honestly, if society deteriorates to the point it can no longer organize a government, a few highly radioactive spots on earth are not going to be a huge deal.
Nuclear waste facilities also have nice economies of scale in the sense that a single site's incremental cost of storing an additional ton of nuclear waste is negligible. So unlike global warming, you're not saddling the future generations up with a massive bill.