A post on Battle.net from Jeff Kaplan explains that Blizzard is working on a way to export Overwatch highlight clips "someday," should the technology become available to permit such a miracle (thanks PCGamesN). Here's word:

The game will temporarily archive 5 highlights from your last play session. Playing only one game will only result in one highlight. Highlights are not currently saved if you close the game and come back. Our hope (and what we're working toward) is to allow players to save highlights or export them to a video format someday.