From experience. Contamination is patchy. Especially there. You can briefly step into some plutonium contaminated patch for a second, and if you had no proper decontamination on exit - bring this shit home. Once its inside the body - there is no cure.
They had all of us go through these scanners on our way out of the city to ensure nothing dangerously radioactive clung to our clothes. They didn't look like the most robust or reliable of devices, though. Found someone else's photo of them: https://www.flickr.com/photos/73155136@N00/518812012
I mean, I don't imagine people going there to be eating random stuff they find. So maybe from dust and other small particles you don't notice?