At about 1630 Alaska time, a Piper PA-31-310 was damaged during a gear-up landing at the Yakataga airstrip. The pilot was not injured. The pilot said he encountered moderate to severe turbulence during his initial approach to the airport and did not think he was going to be able to complete the approach. The turbulence subsided as he turned from base to final, and he then concluded he would be able to land. He said he elected to fly down the turf and gravel-covered runway farther than normal to avoid a pool of standing water on the runway. As he flared the airplane slightly for landing, he felt the cabin step on the right side of the fuselage drag the runway. The pilot said he became preoccupied with dealing with the turbulence and missing the pool of water on the runway, and due to the increased power setting he used to avoid the water, the gear up warning horn did not activate.