Breaking down your tissues begins way before you run out of fat.

Example:
3,500 calories per pound of fat
2,000 calories per day
150lb person with 15% body fat (a normal looking person, almost 'skinny' for an American)

This person has enough fat for 40 days of no eating. If you manage to scrounge up a few roots or berries every day, you could make it two months on eating almost nothing.

Now, this 'thermodynamic approach' is very over-simplified, and doesn't work with humans. But my point is, you're not running out of fat anytime soon.