The amount of trash generated by humans is humungous. The United States alone produced 251 million tons of municipal waste in 2012, and the amount climbs to 208-million-metric tons in 2015. The whole world created 2.6-trillion-pounds of waste in 2012; equivalent to 7000 Empire state buildings.
Assume if the USA decides to send 208 million tonnes of garbage to space at the current cost of $16,000/kilogram. The USA alone would have to spend 208 trillion dollars to carry that weight into space.
The single nuclear reactor produces 23-30 tons of waste. Even if the scientists and engineers figure out a way to lower the cost 16 times ($1000/kilogram), it would cost the USA around 1.6 billion dollars to send that nuclear waste to space. We have 435 nuclear reactors in the world that would make almost 2200-2300 tons of nuclear waste every year. Not to mention the safety requirements nuclear waste requires for the journey, you can just imagine where would that lead us to.
The problem has turned glacial. Here is our favorite part of the problem: Imagine even if we are willing to spend that much amount of money to carry the garbage into space, it would just add to space debris. Humans are not only responsible for polluting the Earth, but space also. According to the tracking mission in July 2015, a total of 17,852 objects has been artificially placed into space by humans.
These objects are trackable due to their significant size. In the debris found in 2013, 170 million pieces were smaller than a centimeter, 670,000 were between the range of 1-10 centimeters, and 29,000 were the large pieces of debris present in orbit. The space debris is already a hazard to the space crafts in space. Now, we wish to launch 60 times more debris, just to keep the USA garbage-free. Unless we are throwing garbage into the Sun, this idea of dumping garbage is just not viable.