Children children children(and people older than me.) Story is vital, there are some games that you can be just dropped into sure but those are all mini games, Tetris, Super Mario bros(original few), Donkey Kong. They have quick text blurbs in an instruction manual or on the side of an arcade that you don't really need to read to understand what's going on.
Other games like Final Fantasy NEED a strong story, it's an absolute must. Try playing through Final Fantasy I on the NES, they just drop you by a town and basically say "good luck!" RPG's Adventures etc need a story to connect the sequences of events or else they can just make it linear and everybody would complain every game is too linear.
Storys open up emotions, connections with the characters, and can even open choices in the game itself(I'm citing Chronotrigger where making some minor choices can affect the games storyline and ending)
This whole thing is moot though because the best story driven game has already been made and remade twice since. Lunar: Silver Star Story a game that proves love stories can fit into games without making them girlie or ruining the pacing.
