Nuclear is the best btw.
Look at it from a risk assessment standpoint.
The barrels will be made.
You can put them in the deep ocean in the Marianas trench, they will degrade immediately and leak quickly. they will be diluted enough to make the leakage relatively safe. most of the upper foot chain doesn't get to eat those creatures down there. But bad optics and enivonmentally unreasonable.
You can surface warehouse them. They can be regulary check for leaks, but can be struck by terrorists, meteors, airplanes, capitalism. Still bad optics.
You can bury them outright. can't check for leaks, they might make it into the water table. bad for animals which might enter the food chain.
You can put them inside a tunnel/vault. You can, until there is an event (and maybe afterward) check for them for leaks. move them to somewhere else, maybe even use them for other things if you find ways to do that.
Nothing is indefinite. But of all the places we can put them, a maintainable underground vault is less likely to get fucked then we as human are at fucking them up.