One important detail is that the funguses are not necessarily radioactive [x]. They absorb the gamma radiation, that it's similar to the X-rays. After a radiography, x-ray or gamma-ray exposure the things don't become radioactive. But too much radiation kills almost anything. This process is used to produce irradiated food (http://en.wikipedia.org/wiki/Food_irradiation) that is safe for eating (when the irradiation is done properly).

[x] The funguses inside Chernovil are surely radioactive because they also absorb the water that is contaminated with radioactive elements. Don't eat them :).