One possibility is that to have a cannon fire the waste into space. Instead of explosives powering the cannon use magnets to do so. The cost of the electricity would be very small, though the cost of the cannon would be high. Plus you would need to go though the atmosphere without burning up, or slowing down much which might be a problem.
1) Even the electricity to fire it would be enough to convert the material to plasma, destroying all organic type threats it might pose.
2) Getting it through the atmosphere is a non-trivial problem. You need a *BIG* gun to accomplish this, the atmosphere will pretty much stop any projectile with a cross sectional density below 14.7 pounds per square inch assuming your gun is at sea level. (Yes, I said cross sectional density, not pressure. It's not a coincidence that this matches Earth's atmospheric pressure, though.) The only way you'll accomplish this is with a long, thin projectile and it will have to be far bigger than the biggest shell ever fired.
3) If you want to hit the sun the velocity after leaving the atmosphere must be 41 km/sec. You'll need substantially more than that upon leaving the barrel. Suppose it tumbles a little bit? It's going to be shredded. The Challenger got torn apart by twisting a bit at a *FAR* lower velocity.