Is working in public hospitals dangerous for your health?

Possibly: The germs in hospitals are more serious and resistant to antibiotics - stay away if you can.

Any hospital may be: Depends on the job, exposure to patients, body fluids, etc. All hospitals have sick patients, lots of germs floating around. Any one working with "sharps"-needles, etc. Has risk of needlestick infection. Be careful, wash your hands often, or use alcohol hand sanitizers, e.g. Purell. Private hospitals have the same germs too. Worried about crime, neighborhoods, etc.? Different issues.

Only in the sense: You can catch a antibiotic resistant super strain bacteria.