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answers:
Raymond say: Earth is orbiting the Sun at 30 km/s
From Earth, the "Solar escape speed" (the speed to escape the Solar system) is around 42 km/s.
This means that for sending a probe outside the Solar system requires a "delta-V" of only 12 km/s (this is what we did for the Voyagers and for New Horizon).
However, to drop a rocket towards the Sun we would need a "delta-V" of 30 km/s, more than twice what we have been able to achieve, so far.
Saturn 5 (or Saturn V, if you remember your Roman numerals)?
It only reached a "delta V" of 11 km/s to go to the Moon. And that was our best "heavy lifter" if you want to sent out tonnes of stuff.
The space shuttle could only reach 8 km/s, and it could be modified to reach 9 or a tiny bit more. However, it could take heavy loads.
Therefore, the answer to your question is no.
We are not yet at the level where we could do that.
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vic say: Good idea we wouldn't need to send it all just the most dangerous stuff o it wouldn't be cost prohibitive for a few loads, it wouldn't be going out of the solar system like someone mentioned it would be going to the sun in the solar system
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Brigalow Bloke say: Do you want to pay for that?
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tham153 say: currently 93% of launches are successful. what about the garbage on the other 7%?
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Roger K say: If you want to pay tens of thousands of dollars every year for garbage collection, sure. It is extremely expensive to launch anything.
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Mutt say: Could we? Yes, it could be possible. But the question is who is going to pay for it?
Taking all the non-degradable garbage and launching it into space is going to take an extremely large rocket ship if you want to get to the Sun. Look at the size of the Saturn V rockets used to launch the astronauts into space to go to the moon. And the garbage is going to weight a hell of a lot more than the CSM/LM with all the supplies, tools, and astronauts. And adjusting to today's dollars, each launch of the Apollo missions with a Saturn V rocket (earlier missions that did not go to the Moon used smaller rockets) was over $1 billion.