Has there ever been a two engine Air Carrier anywhere that has had both engines fail at some point during the takeoff run that resulted in an accident?
The one I can think of that is arguably related: during the takeoff segment of a training touch-and-go aboard an Airbus A320 in 2018, an impressive chain of events led to a failed elevator and early gear retraction, and then both engines hit the runway, ultimately failing in the air, but they didn't fail on the runway. The crew managed to turn back, crash land, and survive.
This one is close, the first failure happened just past VR, second shortly after. The aircraft only made it 15ft off the ground.
On 9/1/2005 a DA-20 Falcon (Business Jet) was taking off from Lorain County Airport in Ohio. Shortly after rotation, the aircraft hit a flock of birds, causing the #1 engine to flame out. As the gear was retracted, they hit another flock, which caused the #2 engine RPM to roll-back. The pilot was not able to sustain airspeed or altitude and crash-landed, sliding through a ditch and airport perimeter fence, crossing a road and ending in a corn field. Aircraft sustained major structural damage beyond economical repairs. Both pilots were taken to hospital. Only the copilot sustained minor injuries. The NTSB investigated, case # IAD05LA129. Costs totaled $1.4 million.
A second source says:
The pilot was flying the airplane at takeoff, as the copilot monitored the flight instruments, and made "V" speed and rotation callouts. When the airplane was about 15 feet above the runway, birds flew up from both sides of the runway, and into both engines. The number two engine "surged," and made several "loud reports" before the copilot noted a complete loss of power on the number two engine instruments. The copilot announced the loss of power, and the pilot called for retraction of the landing gear. The airplane climbed for about 10 seconds, before the copilot observed the gas producer (N1) gauge on the number one engine decay through 50 percent. The stall warning horn sounded, and the pilot adjusted the flight controls for landing. The airplane contacted the runway "straight and level at approximately 130 knots," with the landing gear retracted, about 3,000 feet beyond the point of rotation. The airplane overran the runway, struck a fence, crossed a road, and came to rest in a cornfield about 1,000 feet beyond the initial point of ground contact
There is another business jet example in that wildlife strike PDF where they hit a deer and the thrust reversers failed (part of the engine) so they couldn't stop fast enough and crashed into a ditch past the runway.