The common answer is world events, but those are still so complicated that even their little brothers, instanced events, are still in the stages of infancy. We may be able to clear out a castle in an instance, but can we destroy one? And the very fact that it's instanced means that we can leave, walk back inside the door, and nothing has changed. We chalk it all up to coding right now that the prisoner we just rescued a few minutes ago still remains in his cell, and we simply sigh, resigned to the fact that we're not really changing the world, just leaving it reset for the next group of players.
Still, there have to be some ideas floating around that could work to bring around a great story in a persistent world.
What if, after completing a certain quest, you were transported to a different version of the game's overworld, one in which every character, building and town had moved on? You could go back and visit characters you'd met at the beginning of the game, but they would be completely different (since you're in a different "version" of the overworld). Change the world in this way a few times over the course of leveling (or with each expansion), and you've got the makings of some good storytelling. Sure, you couldn't go "back in time" with your current character, but start a new alt, and boom, you're back in the first world. There are also problems on the creative end, not to mention the population issue (do you have enough artists to create three versions of your overworld, or enough players to fill them all?), but a little tuning could make it work.
The other option is to somehow code different views for each player (as they each hit different points in the storyline), but this gets messy as well. And there's a story to be told in character advancement as well-- right now, most character changes come in the form of new items, but what if your character aged as your /played time grew, and NPCs reacted not just to your race or class, but your age as well, and how long they had known you?
There is a lot of lore and backstory to be had in lots of MMOs, but we are just scratching the surface of actual storytelling in massively multiplayer games. It's not easy to put forth a solid narrative in this format (not without a lot of suspension of disbelief on the part of the player, that is), and yet there's no doubt that the people crafting this strange genre will just get better and better.