There are other risks which won't show up on a geiger counter. For example the ingestion of radioactive material, which will cause a much higher effective dose than estimated by a geiger counter or dosimeter. This is called a committed dose: https://en.wikipedia.org/wiki/Committed_dose

When I worked there, and I spent 2 field seasons in 1993-94 in the 80-120Ci/sq.km zone, we had two proverbs, briefly translated as: "If you want to be a parent, keep you balls contained in lead" and "You can eat apples here, but you should bury stubs at least 140 cm deep". Only the second one was true. Mesocarp of fruits was clean, but seeds were normally contaminated with Cs 137 (replaces potassium in meristema).

Took me a minute to figure out what a stub was. Apple core, for those wondering.

How easy is it to ingest something by accident while in the area?

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?

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