[BurningFrog]

The classic answer is to bury extremely long lived waste in a subduction zone at the bottom of the ocean, such as the Mariana Trench.

A "subduction zone" is where two tectonic plates collide and fold into the depth of the planet. So anything you bury there would naturally travel deeper into the planet over the eons.

I haven't heard any serious arguments against this, but I suspect it's impossible because people think it would "contaminate the oceans".

[fromthestart]

I dont think it is so easy. Subduction zones have what are called accretionary prisms, which are basically sediment that is sloughed off the top of the subducting plate. Thus, if you want the nuclear waste to be subducted, you have to bury it deep enough, though I can't say quite how far - given the scale of the crust, I'd say you're looking at something on the order of miles at least. Which is extra challenging considering subduction zones are already under thousands of feet of ocean.

[pfdietz]

And even if it were subducted, the descending seafloor sediments liberate water and other volatiles once they get deep/hot enough, and this produces volcanism.

[WalterBright]

Extremely long lived radioactive waste will, inevitably, have extremely low levels of radioactivity. I.e. not an issue.