This is simply not true. The longest lived fission products will become stable elements within just a few hundred years. The longest lived fission product has a half life of about 30 years. The total amount of waste material is extremely small. All of the high level waste produced in the United States in the last 30 years could fit onto a single footbal field in a single layer of licensed storage containers.
Most myths have some basis in the real world of facts and observations.
First, let's address the longevity issue. While all isotopes that are produced when uranium splits are relatively short lived, some of the uranium atoms do not fission with the first neutron impact. Instead, they absorb the neutron and become a more massive isotope. Some of the heavier isotopes will fission when hit by the next neutron, others will absorb the next neutron and become even heavier. The process does not go on forever.