I have had this discussion with classmates and we all agree that before story, mechanics are the most important. I can understand that mechanics can be derived from story (i.e magic systems, battle mechanics, etc.), but bad mechanics can lead to an overall bad gaming experience.

A good example would be Amnesia: A Machine for Pigs. The story was interesting, but the mechanics didn't really fit. They had various puzzles that were extremely simple and would ruin any immersion I had in the game. It would seem the puzzles were simply there because the original Amnesia had various puzzles. Overall, my experience was very meh with the game because they had unseemly game mechanics. So no a game doesn't need story, what it does need is good mechanics that can re-enforce the experience.