The pilot reported that the airplane was refueled while he performed the preflight inspection at night. He recalled that the fuel selector had been set to "both," and about halfway through the flight, he noticed that the right fuel tank appeared full but that the left fuel gauge indicated that fuel was being consumed from the left tank. He decided that he would "check the situation " when he reached his destination. About 5 miles from his destination, he noticed that the left-wing tank fuel quantity indicator showed that the left tank had 5 gallons remaining and that the right tank was still full. The pilot stated that, during the approach, the airplane became "very unstable and seemed to be uncoordinated." The stall warning horn sounded, and he applied left aileron and left rudder to align the airplane's heading with the runway heading but was unsuccessful. He aborted the landing, established a climb, and retracted the flaps. The Low Fuel warning light illuminated, and he switched the fuel selector to the right tank. The engine lost power, and he made a right tear drop turn back to the runway. When the airplane was about 1,000 ft above ground level, he pitched for best glide airspeed, about 70 knots, and he attempted to restart the engine to no avail. He lowered the nose, the airspeed increased to 85 knots, and he added 20° of flaps and landed on the runway. The airplane bounced hard and then came to a stop. After the accident, it was determined that the right fuel cap was not secured and had fallen off the airplane onto the runway during takeoff. The airplane sustained substantial damage to the lower fuselage longerons. The pilot reported that there were no preaccident mechanical malfunctions or failures with the airplane that would have precluded normal operation.