I was a volunteer for the rescue program in 2018. I am not sure what the exact criteria were, but dogs from the Chernobyl exclusion zone have been re-homed before.
A few dogs which otherwise would have been good candidates were too internally contaminated to re-home. Not too bad, it would have been something equivalent to a few x-rays a year if the dog slept on your bed every night. But the modal rescue dog has no radiation risk at all to worry about.
Beyond that, the workers in the exclusion zone are quite fond and protective of the stray dogs, so I think there would have been significant blowback if a significant portion of the dogs were re-homed.
A few dogs which otherwise would have been good candidates were too internally contaminated to re-home. Not too bad, it would have been something equivalent to a few x-rays a year if the dog slept on your bed every night. But the modal rescue dog has no radiation risk at all to worry about.
Beyond that, the workers in the exclusion zone are quite fond and protective of the stray dogs, so I think there would have been significant blowback if a significant portion of the dogs were re-homed.