Irradiated foods are bombarded with ionizing alpha and/or beta particles. This causes cell damage to organisms responsible to spoiling or contaminating the food (thus improving safety and shelf life), but does not cause any of the matter to become radioactive itself. In a nuclear accident, there is a chance that a victim becomes exposed to neutron or gamma radiation, both of which can induce arbitrary matter (including in a victim's body) to become radioactive. Whether or not their bodies actually became radioactive sources, it's pretty clear that the medical staff assumed that was the case, considering victims of ARS from the incident were burried in zinc coffins after their deaths.
