The spherical cave would be used more than once.The rubbish would be completely vaporised after each detonation.True,there would be some residual radiation,but this would be deep underground and would be of the same magnitude as encountered in current underground nuclear tests.I would imagine that as the underground cavern would be used over and over again that the costs would start to come down.Besides radiation is constantly seeping from granite rocks from the decay of natural uranium.So radiation from deep underground is nothing new!
There are of course other alternatives,such as:
Landfill: -not in my backyard!
Incineration-problem:no-one wants the by products of incomplete combustion (Dioxin) ending up in the food chain.
Recycling:okay at a local small-scale community level but how do you organise it for large cities such as New York(~14Million)or Tokyo & London?
Of course there is also the dumping at sea option,but I wouldn't like to eat salmon,prawns,mussels,or tuna that grazed close to barge-loads of rubbish.
Hope this gives you food for thought!