For me, this depends on the game genre... FPS? All the story I need is the 'end condition' (basically what do I have to do to win this level), and I can play.

For me, this depends on the game genre... FPS? All the story I need is the 'end condition' (basically what do I have to do to win this level), and I can play. "reach point X", "detonate bomb", "kill everyone".... sure. Strategy games? Same... kill everyone, destroy base, done. But for example, point and click adventures with a shitty story? Totally unplayable for me.