Depends on the game. In my experience, games designed for multiplayer PvP from the get-go never have decent story, and games designed for story usually don't have very good PvP. Because the dev team is focusing on one, or the other, and story in particular lends itself to a single-player experience.
Overwatch was never intended to have in-game story in any major sense. Neither did earlier games in the genre. It's all about the action. You don't need a story to back games like chess or the like, either, because the game itself is what you're there for.