Do people who work in nuclear facilities need hazmat suits? And if so when all times or just when near the reactor? How does it all work?

Not suits, but radiation detectors.

I wore a radiation tag every day to work while at a National Lab that designed, built, and tested nuke weapons. And every so often they'd collect the tag and read the dosage off it to ensure I didn't get overdosed.

The Lab is one mile square and I had to wear it all the time everywhere within that compound. Not just when I was in a building where nuke materials might be stored.