Outside of comic books, when you are exposed to radiation, your DNA is damaged and you get ill. Sometimes very ill: just witness the terrible effects of the radiation released in the Chernobyl nuclear disaster 20 years ago.
But if only life were like it is in the comics! Spiderman got his powers when bitten by an irradiated spider. The supernatural powers of the X-Men are a result of environmental genetic changes. Bruce Banner's green alter-ego, the Incredible Hulk, bursts into life after he is exposed to gamma radiation.
In reality, changes to our DNA -- mutations -- occur randomly in our genome and only rarely have a beneficial effect. This is why natural selection proceeds gradually. Only those rare mutations with beneficial effects, say on our muscle strength or ability to speak, will produce something that natural selection can work on. Most mutations have an effect that is neutral with respect to evolution.