During takeoff, the engine lost power at about 250 ft AGL. At that time, the aircraft was about half way down the 4200 ft runway. The pilot applied carburetor heat and full flaps with the intention of landing on the runway. As the aircraft touched down on the packed, snow covered runway, the engine regained power. The pilot reported that there was insufficient runway remaining to stop the aircraft without striking a hard packed, 10 ft snowbank. Therefore, he elected to go-around. During climb-out, the engine lost power again, and the pilot was unable to return to the runway. The aircraft crash landed in a snow covered clearing near a dump on wooded, hilly terrain. An investigation revealed no precrash discrepancies except that the annual inspection was overdue. The engine ran satisfactorily during an operational check. The carburetor was disassembled and checked for contamination. A very small piece of ice was found. It was noted that water may have entered due to melting snow in the impact area.