The problem with garbage disposal in space is that nothing every really goes away. It drifts along towards the nearest object that has a gravitational field and starts orbiting around it. After years of launching rockets, doing spacewalks, building space stations and installing satellites, we've created something of a garbage problem around the Earth.
We currently estimate that there are 8,000 individual bits of 'space junk' floating around up there, although that number grows all the time, and the more there is, the more of a hazard it poses. Once in full orbit, the junk whips around the world at 17,000 miles per hour - fast enough for it to completely destroy a television or radio satellite if it hit one. The debris is mostly made up of older, dead satellites, along with booster rockets and leftover detritus from space station construction - there are even astronauts' dropped tools and gloves up there. Discussions about what to do about it, and how to clear it before it forms a mesh that endangers future space missions, are ongoing.