When you run out of HP, you must save or die.
Unless you're a Barbarian, that is. As the Constitution class, Barbarians can withstand the first death blow. They can keep moving and fighting at 0 HP. It's only if they're hit again while on 0 HP that they fall into Save Or Die territory.
This means Barbarians can be crazy reckless - charging through doors, jumping down pits, and checking for traps with their face. When the chips are down, the Barbarian will be the last line of defense - arrows through his chest, an axe hanging off his shoulder, somehow still alive and roaring.
As they gain levels, they can get feats like double attacks on the edge of death, and the ability to stay up as long as they keep passing their death save.
Anyway, that's what I'm going to do with them in this RPG I'm making. I've decided to call it: