The aircraft was on a round trip flight from Barrow, AK to Point Hope, AK and return. After landing at Point Hope, there was a 20 min delay while waiting for the ground crew. The pilot stated that he ran the engines for some 10 min or so due to the high winds and cold. He stated that the wind was averaging about 45 kts and was too strong to maintain control on top of the aircraft for refueling. The pilot calculated that he had more than enough fuel to return to Barrow. While returning, he flew past two possible refuel points and a third place where he could have landed. While over water near his destination, the left engine flamed out and the pilot incorrectly shut down the right engine. He was able to restart the right engine, but it soon flamed out also. A successful forced landing was then made on the tundra. According to the pilot, the engine flamed out due to lack of fuel.