What you have to worry about is not whether or not something has been exposed to radiation, but whether or not it is radioactive.
Radiation includes alpha (high energy helium nuclei), beta (high energy electrons), and gamma (high energy light). If I were exposed to radiation, I might suffer several consequences depending on the intensity of exposure. These forms of radiation, however, only exist for a fraction of a second after creation. If a fish is exposed to gamma rays, and I eat the fish, then I am not exposed to gamma rays, as the gamma rays no longer exist.
The true cause of radiation sickness would likely be when a person consumes the radioactive source itself. If the food contains radioactive cesium, and I eat the fish, then I am exposed to the radiation. Why? Because the radiation is actually created inside of me.
Hopefully this clarifies some stuff, because many people think that “radiation” is a physical object, but in reality any single particle of radiation only exists for a fraction of a second (and moves close to the speed of light).