Re: How is this waste?
Well said, sir!
U-235 has a half life of 700 million years, and U-238 of 4500 my. They are not the problem; it is fission products that are.
I believe, though I would like to see someone else do it first, that one can safely walk past a freshly manufactured fuel rod. But a rod just withdrawn from a reactor is spitting wild rates of radiation from isotopes with a half life of just a few minutes, and would probably kill you. So those rods are left in a deep pool of water for a year, until there are two main fission products left: caesium-137 and strontium-90. These have half lives of about 30 years.
A half life of 30 years makes them emit, weight for weight, over 50 times more radiation than the classical radium (half life ca 1700 years). They could, however, be useful for peaceful applications of radiation. This may be why the story mentions hospital uses. Or, as commented above, they can be turned into glass and buried.
I have read that much of the earth's internal heat comes from the decay of potassium-40 in the earth's crust, with a half life of many billions of years. So I calculated the internal temperature of a football-sized sphere of sodium strontium silicate (glass). It would be red hot at the centre. So burying glass needs some thought and design.
But the story is unconvincing, a cloud of ackamarackus to hide some real objective, which may even be as mundane as money.