According to the U.S. Environmental Protection Agency, the disposal of radioactive waste varies by type, but generally it must be stored somewhere until radioactive decay renders it safe. Low-level radioactive waste like contaminated safety gear can be safe enough to place in traditional landfills, while high-level waste like spent fuel generally requires special storage for hundreds or even thousands of years to prevent radioactive contamination.
In addition to storage, some radioactive wastes can be reprocessed and reused. Wikipedia explains that some types of spent fuel can be purified, removing the fission products, so that it can be reused in new reactor types. With reprocessing taken into account, the amount of spent nuclear fuel that requires long-term storage is relatively small. According to the Nuclear Energy Institute, the spent fuel from 50 years of nuclear power generation can cover a single football field to the depth of 10 yards.
The length of time radioactive waste must be stored varies depending on the individual isotopes and actinides involved. Some materials have a relatively short half-life and decay quickly, but release enormous amounts of radiation in the process. Other materials have a much longer half-life, and they can remain radioactive for millions of years; but they emit such a low amount of radioactivity that they are technically no more dangerous than the Earth's background radiation.