You don't immediately die from exposure to radiation though.
The Union of Concerned Scientists estimates between 4,000 and 27,000 people died as a result of the disaster, where as Greenpeach places the figure much higher at between 93,000 and 200,000.
Then you have the problem of how you dispose of Nuclear waste without it impacting on the environment.
It takes a lot of radiation to kill you, and it is very difficult to determine if an exposure has "pushed you over the edge" so to speak, and you get cancer, so I am not sure if these numbers can really be known with any certainty.
Greenpeace is an absolute disgrace for their anti-nuclear stance, they are more interested in environmental PR then the actual environment.
Far more people likely die from exposure to any one of the thousands of chemicals that are everywhere, in our food, all our products, even the construction materials that we use to build our houses. I worry about that funny smell when I cycle past the chemical plant on my way to work, and when I get stuck in traffic behind some stinky diesel. Many things that we think of as "safe" are like you said above, things that don't kill you right away, but rather add up over time.
There are thousands of toxic waste sites from various industrial processes, mines, chemical plants, factories, dumps, and forget half-life, some of them never break down, the so-called "forever chemicals". Slowly they are all leaching into the environment, mostly through groundwater. And like Chernobyl, it is very tough to prove they were the cause of one's cancer when it comes.
The media has managed to convince the public that radiation is some kind of super poison, when radiation is all around us, and always will be. The other point that seems to escape discussion is that radiation is super easy to detect, even at background levels. This means that if air or water in an area become polluted for some reason, we know it, and can do something about it. This is mostly not the case for all the other industrial nasties that we are exposed to.
To put things in perspective, the amount of radiation released into the environment from the atomic testing in the 50s and 60s is estimated to be between 100-1000 times that of Chernobyl.
Like I said previously the idea of "nuclear waste" is becoming outdated. Today's reactors can run on re-processed fuel from the previous generation, in other words they can burn nuclear waste. Only a few percent of the possible energy is extracted from the fuel, so it is not really "waste" per se.