Storing nuclear waste below ground already is a thing. But drilling down would be a problem. It's inherently unstable to generate heat. We need to store it somewhere a) stable b) low heat (as we don't want a meltdown) c) where any storage vessel won't melt from environmental heat (so not too deep due to temp gradient or deep but in cold rock d) where it won't interact, if there are issues, that would affect groundwater.

We often use salt mines for this as salt is very good at reducing heat in the source, low permeability and also self heals. It's pretty ideal for storage of nuclear waste. A borehole wouldn't suit for all the reasons that a salt mine does.