Radioactivity in foods can occur by two routes:
The irradiation process involves passing food through an irradiation field. While in the irradiation field, the food itself never contacts a radioactive substance. In addition, the ionizing radiation used by irradiators is not of sufficient energy to disintegrate the nucleus of even one atom of a food molecule.
The U. S. Department of Agriculture has documented [local] the risk analysis issues associated with irradiation of meat.