The most frustrating loss by a team in NFL history may have been the Saints versus the Dallas Cowboys on September 25, 1983. In that game , Kenny Stabler was QB for the Saints, and Danny White was his opposite number for the Cowboys.
The Saints led 20-13 with around 10 minutes left in the 4th quarter. The Saints drove deep into Dallas territory, where the drive stalled. They called on Morten Andersen to kick a FG and extend the lead to 10 points. The FG attempt was blocked, and a Cowboys defender picked up the ball and returned it 62 yards for a TD. 20-19 Saints. Then the Cowboys missed the extra point, and the score remained 20-19. The Cowboys kicked off to the Saints, who failed to move the ball, and had to punt. The Cowboys took the ball over around their own 35 yard line, and drove down deep into Saints' territory around the 2 minute warning. On third down, from the Saints' 4 yard line, Danny White threw a pass into the end zone, which was picked off by LB Dennis "Dirt" Winston. Winston tried to run it out of the end zone, but was tackled on the 2-yard line. On the next play, Stabler was tackled in the end zone for a safety, and the Saints lost 21-20.
That was probably the strangest way to lose a game I can think of, probably stranger than the River City Relay game in Jacksonville.