There are three major types of exposure to radiation:
- irradiation
- activation
- contamination
By "irradiation," I mean that the exposed material is actually absorbing energy from some radioactive source. Generally this energy is absorbed as fast charged particles (alphas or betas) lose energy by creating electron-ion pairs in neutral matter. (Gamma rays, which are high-energy photons, aren't charged themselves, but also can knock electrons free from their host atoms or molecules.) It's the changes in the ionization states that make irradiation dangerous for a biological system, since biology depends strongly on the details of electrochemistry.
If your radioactive source is sufficiently energetic, your ionizing radiation can lose energy by exciting the nuclei of neutral matter, in addition to interacting with the electrons. For energies above 8-10 MeV, you can start to liberate neutrons from stable nuclei. The neutrons then wander around until they find another nucleus to capture them. This generally makes the "donor" nucleus and/or the "recipient" nucleus unstable against beta-decay, leading to further irradiation later. An example you've heard of is that neutron capture on nitrogen-14 tends to eject a proton,
$$
\rm n + {}^{14}N \to {}^{14}C + p,
$$
and the carbon-14 is unstable with a lifetime of a few thousand years. That particular process happens in the atmosphere, which is why currently-living things have a particular concentration of carbon-14 which fades away after they die, but there are similar processes when neutrons interact with other elements. This is known as "activation," because previously-stable materials become radioactive when irradiated by neutrons. Neutron activation is the reason why non-fuel materials from the innards of a nuclear reactor or an accelerator are dangerous.
"Contamination" is the exposure process that is easiest to understand. That's where you have some radioactive material (whether naturally radioactive, or activated as above) which crumbles into dust, and the dust gets into places that it shouldn't and gets carried away. It's concern about contamination that makes radiation workers wear those disposable Tyvek bunny-suits. The suits aren't any better at shielding against irradiation than your skin or your own clothes. But if the suits get contaminated, it doesn't matter because you throw them away when you're finished with your job.
The most common form of irradiation in medical settings is exposure to gamma rays from cobalt-60. That's a source that can be made completely sealed, so that only the gamma rays can escape. There's not enough energy in the gamma rays to drive neutron activation, and unless the source is cracked or damaged there is no risk of contamination. Irradiation by cobalt is like illumination by light: when you turn off the lights in your bedroom, generally the walls go dark immediately.
I have some biology questions about the plot your describe, but that's beyond the scope of this answer.
