I absolutely loathe and hate the "depends on..." kind of answers, but its the only answer that is appropriate here
A platformer needs no story, i dont give a flying fuck why Mario or Crash or Sonic is going to World ____ to do ____ and defeat ____. Its completely pointless to even pretend there is a story. I cringed so much during story bits in Galaxy 1 and 2, like really? Are you really trying to have a moment here?
But in a game like Silent Hill 1-4 it is crucial, it is in fact the sole purpose to play the game for me. The narrative ties into the actual game, the art style and the things the player feels during his or her adventure.
Same with MGS, the story is what had me coming back, despite questionable controls and dodgy shooting mechanics.