The private pilot in the light-sport-experimental airplane reported that the flight was conducted with the intent of burning off fuel, because he planned to pick up a passenger. When the pilot entered the downwind traffic pattern, the engine stopped. The pilot reported that, "my mistake had been to not switch fuel tanks before the one I was on was sucked dry." The pilot switched the fuel selector to a full tank and attempted to re-start the engine as he maneuvered to make the runway. He was unable to re-start the engine and he over shot the runway. The airplane's airspeed was slow and the airplane stalled. The airplane landed hard on the left side of the runway centerline and remained on the runway when it came to rest. The airplane sustained substantial damage to the firewall and the fuselage aft of the firewall. The pilot reported that there were no preaccident mechanical malfunctions or failures with the airplane that would have precluded normal operation.