There are fundamental mistakes here. First, thinking of it as waste is a mistake. And thinking of getting rid of it is a mistake.
The very fact that this material is radioactive tells you there is energy available in it. Especially in the stuff like Pu. Just because, right now, we don't have a fuel cycle that uses it does not mean we will not develop one.
The thing to do is put it someplace it won't cause problems, but that we can get it back easily if we want it.
Yucca Mountain would be a very good choice.
https://en.wikipedia.org/wiki/Yucca_Mountain_nuclear_waste_repository
Sadly, this is so mired in political in-fighting it seems unlikely it will ever get put into operation. When a thing such as this gets brought into the legislature and the courts it seldom emerges intact.
But the idea would be to put this material in a place that is suitably stable for about 100 years. You want someplace with little ground flow of water and some solid rock formations. You leave it there until the shorter lived stuff decays. This leaves you a much easier problem, regardless of the long-term plans.
Then if it is decided to give it a longer time scale disposal, you can make that choice. Or if there is some other plan like reprocessing, then that can be done with lower levels of radioactive material. Or if some other fuel cycle is developed you can feed it into that.
Yucca Mountain should be easily qualified to 1000 years. And with some minor adjustments, to 10,000. You could back fill with concrete and bentonite to reduce ground water flow. You do that to a depth of about 1km. Then anybody who is able to dig that deep will almost certainly have the technical ability to deal with something as minor as 1000-year-old nuclear waste.
But notions of disposing of it by dropping it in a deep ocean trench, or some other difficult-to-retrieve option, are mistaken. You can't monitor it in that trench. You can't get it back and fix the containers if they start to corrode. You can't extract some isotope that suddenly becomes valuable. And you can't feed it into some new fuel cycle.
Plus, such long-term disposal options are likely to be at least as expensive as Yucca Mountain.