What on earth are you talking about?
The most common byproduct of most reactors in use today is Plutonium 239[1]. It has a half-life of 24,000 years. This means it generates extremely high levels of radiation for thousands of years.
While it is true that low levels of radiation aren't very dangerous, high doses are. For example, page 15 of [2] discusses studies on workers exposed to high levels of radiation from Plutonium. There are two studies, which disagree about the dangers of low level (one shows a linear relation, the other doesn't show any at low levels). At high levels, both show much higher death rates than control groups.
That's only Plutonium 239. There are other high level nuclear waste products which last much longer, but aren't produced in such high amounts.
[1] http://www.world-nuclear.org/information-library/nuclear-fue...
This 24,000 year half life of Plutonium 239 is much shorter than something like the 4.5 billion year half-life of Uranium 238. There are things with much shorter half-lives too, of course.
The danger from radioactive substances comes from many things. In Plutonium 239's case the danger is if any gets within the body. The alpha rays it generates don't penetrate through a body, but on the other hand if plutonium particles are ingested or breathed they cause "somewhere between 10 and 1,000 times more chromosomal damage than beta or gamma rays"[1]
This is one of the reasons why the OP is so completely, utterly wrong. Without proper - very long term - protected disposal of high level nuclear waste it kills very easily.
[1] http://www.livescience.com/33127-plutonium-more-dangerous-ur...
"During the Manhattan Project in 1944 and 1945, 26 men accidentally ingested plutonium in quantities that far exceeded what is now considered to be a lethal dose. Since there has been a consistent interest in the health effects of this brand new substance (first discovered by Glenn Seaborg’s team at the University of California in 1940), these men were closely tracked for medical studies.
Forty Years Later As of 1987, more than four decades later, only four of the workers had died and only one death was caused by cancer. The expected number of deaths in a random sample of men the age of those in the group is 10. The expected number of deaths from cancer in a similar group is between two and three."
https://atomicinsights.com/how-deadly-plutonium/
Of course that doesn't count the demon core.
Nuclear scientist have been advocating managed storage of 50 to 100 years instead for quite some time, because there is a chance that (a) we might figure out a better way to separate and then transmute the long lasting fission products to quicker decaying elements, (b) we might come up with better ways to safely store, or (c) we may want to mine the waste for reprocessing the useful uranium and plutonium.
Storing the stuff permanently under ground may not be perfect solution, but to my knowledge it is the best one we have (not producing the waste is not an option, as it has been already done). The plans also take into account that the waste can be recovered from the storage if we later figure out a better way to handle it.
People working on the sarcofagus actually live in Chernobyl town for two weeks on, two weeks off. That way your body can get rid of the built up radiation.
The flora and fauna in the Exclusion Zone is thriving. Radiation levels have dropped significantly, and most animals don't live to the age that they might see consequences of the elevated background radiation.
That's interesting, do you think the radioactive material helped moss to grow, or that the moss patches catch the radioactive material flying around?