Why does radiation sickness cause hair to fall out?
Radiation poisoning causes mutations in DNA that affect normal cell function, often causing them to die.  Cells normally have a number of repair mechanisms but if the damage is too great they won't be able to do so.  In particular, cells that are dividing quickly will not have time to repair their DNA before division and so die far quicker than other cells.  Chemotherapy is built on this concept.  By selectively introducing drugs or radiation that target fast-dividing cells, the cancer cells, which are very dividing rapidly and unchecked, are killed off before most normal cells.

However, there are some cells in the body that are very rapidly dividing.  Hair cells are one such type, hence hair will often fall out during chemotherapy or radiation sickness.  Another big cell type are those in the gut, which are constantly dividing.  That's why you often see people in movies who have been exposed to radiation vomiting.  The radiation targets those cells first, hence those are the first symptoms.