THE PILOT RECEIVED A WEATHER BRIEFING BEFORE TAKEOFF. AT THAT TIME, THE WEATHER AT THE DESTINATION WAS IMPROVING, BUT WAS STILL MARGINAL VFR. THE PILOT TOOK OFF AND PROCEEDED ON A VFR FLIGHT PLAN. WHILE EN ROUTE, HE DETERMINED THAT EITHER HIS COMPASS OR DIRECTIONAL GYRO HAD A PROBLEM SINCE THE GYRO HAD TO BE RESET CONTINUOUSLY. AS THE WEATHER DETERIORATED, THE PILOT BECAME UNCERTAIN OF HIS POSITION. SEVERAL ATTEMPTS WERE MADE TO IDENTIFY THE PLANE'S POSITION WITH THE USE OF A COAST GUARD RESCUE AIRCRAFT. WHEN HIS FUEL SUPPLY BECAME CRITICAL, THE PILOT ELECTED TO MAKE A PRECAUTIONARY LANDING ON THE BEACH OF AN ISLAND. AFTER TOUCHDOWN, HE HELD THE NOSE OFF AS LONG AS POSSIBLE. HOWEVER, WHEN THE NOSEWHEEL TOUCHED DOWN, IT SHEARED OFF. THE NOSE GEAR FORK DUG INTO THE SOFT, WET TERRAIN AND THE PLANE NOSED OVER. THE COAST GUARD LOCATED THE DOWNED AIRCRAFT AND AIRLIFTED THE PILOT AND PASSENGER TO KODIAK, AK.