Yes, but you need to have a writer there from the get-go, and you have to make storylines work within the context of one on one and two on two fights. Most writers are avoidant of this, and most fighting games as a result have ridiculous stories attempting to explain it all. If I had to pick a game that would have a good story for a fighting game, I'd say Karateka. Same goes for Infinite Blade. They aren't necessarily street fighter, but are effectively fighting games in their own right.
Yes its possible, but no on plays fighting games for stories. No one outside of Namco has expanded their fighting game introductions into a proper story like Soul Calibur. King of Fighters tries to, but they only do it to introduce a new batch of characters to the roster.
I like what I do now, which is play SC when I just want to kick a lot of ass and then go to an RPG (admittedly real-time fighting, and lately it's all been platformers like Assassin's Creed and Prototype) when I want a story.