Overwatch may not have a campaign, but it sure has a story. Framed around a conflict called the "Omnic [sentient robot] Crisis," Overwatch was a peacekeeping organization that was thrown into turmoil and subsequently shut down as a result of an internal conflict. The game actually takes place years later, with heroes and villains from both sides waging war against one another.
If you're interested in individual hero backgrounds, you can read them all at your leisure here. For everything else, check out the list below, which will be updated as more come. Even though it hasn't even been properly released yet (that's on May 24), I'm fairly invested in its world.