“Our hope (and what we’re working toward) is to allow players to save highlights or export them to a video format someday,” game director Jeff Kaplan said on Overwatch’s official forums.
Overwatch automatically saves short clips, called highlights, at the end of each game.
Better yet, you should be able to share or export them them right from the game.
Highlights are fun to look at after a night of playing Overwatch, but they vanish as soon as you close the game.