Isotopes of plutonium have a half-life up to 25,000 years, and some elements have half-life into millions of years, So, no, nuclear waste will not become ambient.
And from a practical point of view 10,000 is the same as 1 mil. You can't keep it stored in some facility, you have bury it deep under ground.
Reprocessing/recycling can only reduce amount of waste it does not eliminate it.
Nuclear waste IS a problem.
With proper reprocessing
Plutonium is fuel. It should be going back to the fuel plant, not to the waste pile. Thus the half-life is completely irrelevant.
And the 10,000 is quite relevant as you don't need nearly the engineering to keep it safe for 10,000 years that you need to keep it safe for a million.
Two simple solutions that meet the 10,000 requirement:
1) An old salt mine without nearby bodies of water. (I'm thinking of the disaster where someone drilling in a lake ended up draining the lake into a salt mine underneath.)
2) A muddy location on the abyssal plains away from seismic zones. Make sure your canisters are heavy enough that they'll sink into the mud. Once it's buried transport is very slow, when the container eventually fails the stuff disperses very slowly.
Note, also, that the leakage even from a failure long before the 10,000 years is going to release very little radioactivity.
And while you're at it, note that over the very long run nuclear power
reduces radiation exposure. When we mine and burn up the uranium it won't be releasing radon.