What is radioactivity?
Our skin provide adequate protection against radiation in most circumstances. The problem is when we ingest radioactive atoms.
Many heavy radioactive nuclei decay to form daughter nuclei which are themselves radioactive, and which decay to form grand daughter nuclei which are themselves radioactive, and which decay… forming what we call decay chains. Since some of the nuclei along the chain have pretty long half-lives, it takes decades for the decay chain to complete (finally arriving at a stable nucleus). This is the main drawback of energy production using uranium. The storage and safe keeping of the used uranium fuel is a pain in the ***. This is also why major accidents at nuclei plants (think Chernobyl, Fukushima) are so feared.
Walter Fendt Radioactive Decay Chains Applet: Click HERE.