Is there a way to regain health without spending money or stim packs?

drink water, sleep,

Drink or eat just about anything.

Sleep in an unowned bed for at least two hours.

This gives you radiation poisoning though doesn't it?

Right. Which is part of the game balance.

You need to judge how much you need the health (to save on stimpacks) and how desperate you are at the moment.

Food and drink is cheaper, easier to find, and a quick alternative to stimpaks, but the downside is if you keep doing it you'll eventually (if you start getting into advanced radiation stages) have to find some radaway to flush the radiation out or a doctor to do it for you.

Mind you, there are perks and the like which help to decrease the threat of the radiation as well.

well just a little, you just have to be careful and check levels, but when you are in a pickle water fountains become extremely useful

are there any unowned beds around in the early game?

beds are all over, like there's one in the supermarket, you just have to look carefully, sometimes there may be dead bodies, or those dismembered corpses the raiders leave all over