Nuclear waste shouldn't contain plutonium. That stuff is expensive, so there's an economic incentive, as well as environmental, to reprocess waste and recover any remaining fuel.
Chernobyl (and Fukushima) are entirely different (and much worse) because you do have the nuclear fuel itself being exposed to the environment. But waste fission products in a modern reactor are far less dangerous compared to the fuel that goes in, especially once they have been properly reprocessed and then stored for while.
Isotopes with half lives in the millions of years, as opposed to years or decades, really aren't all that dangerous in terms of radioactivity. Longer half life generally means less radiation.
Chernobyl (and Fukushima) are entirely different (and much worse) because you do have the nuclear fuel itself being exposed to the environment. But waste fission products in a modern reactor are far less dangerous compared to the fuel that goes in, especially once they have been properly reprocessed and then stored for while.
Isotopes with half lives in the millions of years, as opposed to years or decades, really aren't all that dangerous in terms of radioactivity. Longer half life generally means less radiation.