Species that have much lower lifespans than humans are less likely to develop cancer in their lifetimes. These species particularly benefit from the lack of humans around Chernobyl more than they suffer from the high radiation.
Animals are a lot more tolerant of death and mayhem, too. For example, if radiation caused e.g. a 10% incidence of horrible birth defects, a lot of animal populations could survive that just fine, but humans would be horrified at the very idea.
We "deal with it" in the sense that we can survive and even thrive in such an environment. But we don't "deal with it just fine" in terms of accepting it when we could easily change it by e.g. avoiding a particular area.