Player Characters hitting zero hit points and dying is a very grim possibility for all those beloved 1st level characters. We've all been through it and when it happened, we simply rolled up a new character and rolled on.
Everybody has their own opinions and house-rules for death and dying and these are the ones that I'm currently using in my Swords & Wizardry campaign.
How I work character death is 0 hp is unconscious. The character bleeds out per round until they hit a negative number equal to their level + CON bonus (if any) to a maximum of -10. Then they're dead. It gives the PCs a little bit more of a chance while still leaving death as a possibility. That's how I'm playing it now, we'll see how it goes.