The action man had no qualms about plans to watch a sunset from the top of the Burj Al Arab hotel in Dubai - but he started sweating as he reached the high-rise's summit.
He tells Playboy magazine, "When I was doing Syriana, they arranged for my wife, Lucy, and me to go up to watch the sunset over the Arabian Peninsula from the top of that seven-star Dubai hotel that's shaped like a sail.
"So we go to the very top - 60 storeys or something - we're given champagne, and we go, 'Oh my God, this is great'. But as I started to walk towards the edge, my legs locked up. I was absolutely frozen. I completely jumped my neocortex and went straight to this primal, full lizard-brain fear state... I was about to collapse from fear. She (wife) thought it was hysterical."
And now just the thought of fellow action man Tom Cruise leaping from window to window on Dubai's tallest building in the world, the Burj Khalifa, in Mission: Impossible - Ghost Protocol, gives Damon the sweats.
He adds, "Seeing Tom running down the side of that building was the most incredible thing. I went, 'OK, you win. You are the greatest of all time.'
"He didn't just appear to do it for the movie; he actually did that. I, literally, could never do that."