The flight instructor and pilot-receiving-instruction were conducting a cross-country flight in the twin-engine turboprop airplane. According to the flight instructor, while in the airport traffic pattern at the destination airport, the pilot-receiving-instruction allowed the airplane to decelerate below a normal airspeed for the turn from base leg to final approach. He told the pilot-receiving-instruction to increase engine power to accelerate, but the airplane began to yaw left after the engine throttles were increased. The flight instructor “suspected that the left engine flamed out due to fuel starvation” and took control of the airplane. He retracted the landing gear and flaps, applied full right rudder, and rolled the airplane to the right into the operating right engine. The flight instructor did not feather the left propeller because he believed the left engine was still producing limited thrust. The airplane continued in the left turn despite the flight instructor’s corrective actions. The fight instructor did not attempt to turn the airplane back toward the runway because he believed the airplane was accelerating and would clear the hangar ahead of them. The airplane impacted the hangar and came to rest on the roof. The airplane sustained substantial damage to the lower fuselage, both wings, and the right engine separated from the mounts. There was no evidence of a fuel spill in, on, or around the hangar. Examination of the airplane fuel tanks confirmed they were not breached during the accident. When drained, the left- and right-wing fuel tanks contained about 3 fluid ounces and about 30 fluid ounces of fuel, respectively. Although the flight instructor and pilot-receiving-instruction provided different accounts of how much fuel was present before the flight, both pilots stated that the indicated fuel level was within the yellow caution band on both fuel tank gauges before departure. According to the Beech E90 airplane flight manual, takeoffs are prohibited when the fuel tank gauge indications are within the yellow caution band (0 to 260 lbs). Ultimately, the exact amount of fuel available before the flight could not be determined with any certainty. The flight instructor estimated that the 16-minute flight required between 135 to 200 lbs of fuel. The flight instructor opined that the fuel tank gauges were inaccurate which resulted in fuel exhaustion to the left engine. Review of ADS-B flight track data revealed the airplane decelerated below minimum controllable airspeed (VMC) during the turn from base leg to final approach. The airplane continued to decelerate during the go-around. The flight instructor stated that the accident could have been prevented had the airplane’s best single-engine rate-of-climb airspeed (VYSE) been maintained until the runway and landing were ensured. Based on the amount of fuel recovered after the accident, there was likely insufficient fuel remaining to perform a go-around and reenter the traffic pattern for landing with one engine operating.