End a time loop (Alive) after having taken (and healed) over 1000 damage.
This is according to the Achievement Description found on the xbox.com site.
I got this by doing the following:
1. Wake up
2. Acquire launch codes if you don't already have them
3. Get into your ship and put on your suit
4. Jump back down to the first campfire where you wake up (don't die on the descent)
5. Step into the fire until your suit says Vitals Critical
6. Roast and eat 2 Perfect Marshmallows (aka don't let them catch fire) to restore health or as others have suggested, enter your ship and use the medkit found on the wall to the left of your suit
7. Repeat steps 5 and 6 ~11 times to be safe (assuming your character has 100 health)
8. Play out the rest of your time loop (a time loop is 22 minutes total) without dying, waiting for the Sun to supernova