This is mostly self explanatory but I'll provide a couple of pointers for obtaining this.
This can be started through the Endurance Mode, and requires the Endurance Mode DLC. This can be obtained in single player or Co-op.
Any difficulty is allowed, I suggest Adventurer, the easiest.
Namely, one question I had was, you have to COMPLETE 10 days, ie reach the 11th day before extracting.
Helpful cards include cards that give warmth and food for killing things and either infinite fire or poison arrows (I had equal success with both).
Look for and pick up a lantern (usually near a settlement or basecamp) and keep this with you for guaranteed warmth at all times. Once you have found your lantern, I recommend that you pick a tomb and stay in it with your lantern, venturing out only to hunt for food. You need to hunt ~2x per day in order to not starve, otherwise, you can hang out, care free at the entrance. (Credit to Warrz98 for his solution to Team Survival).
This can be stacked with other single player challenges such as Nature Retreat/Vacation/Happy Camper (extract after 7 or 5 days, respectively) or Co-op challenges such as Master and Apprentice (10 days w/ Artifacts), or Bonding Time (Survive 7 days), etc.