Most of the time after an engagement, I am left with very little health. It doesn't take very much for the next person to come along and murder me. Is there any way I can heal up so I can last longer than one enemy?
Like a lot of other games, your health will slowly regenerate over time.
The main difference with Chivalry is that it will only do so when you are, in my experience, standing still, with full stamina. The second you start moving, your health will (occasionally) stop regenerating. It is unclear whether or not this is working as planned, or if it indeed a bug.
So, if you find yourself hurting, find a nice safe place to... do absolutely nothing.
Additionally, per Lyrion's comments, as well as this forum post, health regeneration can be a bit... wonky; sometimes regenerating when it shouldn't. Specifically, in the middle of a fight.
Regardless of the potentially buggy-ness, all sources I've found say the same thing: Your health will regenerate on its own over time. So stay alive!
AFAIK, your health regenerates if you have full stamina and you are not sprinting (walking around still regenerates your health). Also, you won't regenerate health if you are blocking or parrying or kicking or attacking or shooting arrows with a bow (or aiming with a javelin).
It seems to me (after the recent update they sent out), that the wonkiness in health regeneration is mostly solved...