Launching anything just to orbit is expensive per unit of payload mass. "Tyranny of the rocket equation" is the phrase used. As payload mass increases, the rocket gets heavier, so needs more fuel for the same delta-v, but the weight of the extra fuel means you need to carry more fuel... the result is a logarithmic increase in fuel requirements as weight increases. End result: moving large masses into space gets very, very expensive.
Of course, we can't just move it into space. Low earth orbit is a terrible place to store thousands of tons of radioactive material. We want it to escape earth so this crap isn't floating around just waiting to be a problem for someone. So we need more delta-v to get it to escape earth's gravity. This puts the waste in a solar orbit. That's better... but it's still going to float out there forever and we'd probably have some responsibility to keep track of it. If you just barely leave earth behind, the ultimate orbit for the waste will still be pretty close to earth's. You could put it in a stable orbit out a ways, maybe between Earth and Mars, but that increases the thrust requirement further.
People sometimes suggest just firing it to the sun. You're actually better off just accelerating to solar escape velocity and leaving the solar system forever. (it's actually MORE thrust to hit the sun than it is to leave solar orbit behind entirely)
Bottom line, the proposition is incredibly expensive.
There's another problem, though. A percentage of rockets launched will just freaking explode. Rockets going to space are essentially enormous bombs, but we control the explosion just enough to not die when we're riding on them. Launch enough rockets full of nuclear waste, and eventually you've just accidentally created a dirty bomb.
