You will exile just one of them.
When you would lose the game, both Archangels try to replace "lose the game" with "exile this and reset your life total". Since multiple replacement effects are trying to apply at the same time, you choose which one applies first. Regardless of which Angel you choose to use, once you've made your choice, the other Angel will no longer apply, because you're no longer about to lose the game. You'll exile the one angel, reset your life total, and then continue on your merry way.
Also, as was stated in another answer, this is explicitly called out in a Gatherer ruling as well:
If you control two Exquisite Archangels, you choose which one’s effect applies. The other’s effect won’t be applicable after that until the next time you would lose the game.