Your health will start to drop only if your satiety bar equals zero. Satiety bar is marked by green color and is situated in your inventory underneath the health bar. There are no on-screen radiation indicators, you will hear crackling only if you have Geiger counter in inventory. In case you don’t have enough money to buy one, it is recommended to drink alcohol-containing fluid, which will lead to fall of radiation level, and then watch if health still goes down. You can measure radiation level with your Geiger counter (dosimeter).
It is most likely that your character is irradiated.
