I completely agree with Erickson and take it as fact that current MMORPGs lack any sort of overarching story for the player to participate in (even WoW which, despite having an overarching story, rarely let's the player participate in it).
That said, we have to wonder whether it is even possible to do so. Can you make a story that spans hundreds of hours of play? Can you have it involve everybody in the game, even if all of those people aren't necessarily working through it? Can each player actually make a meaningful contribution to that story? Can it still be fun and interesting at the same time?
I think it could be done because we've seen glimpses of it. In Ultima Online, the cities and stores were actually run by the players (to the best of my knowledge; I never played that game). When Lord British fell (even though it was an accident) it was incorporated into the story. In World of Warcraft, it takes the combined efforts of each faction to unlock new areas. In Warhammer Online and World of Warcraft, factions can take control of areas of the map, affecting what the other factions can do there. All of these are small glimpses of the kind of persistent storyline that current MMORPGs miss out on.
I think part of the problem is that MMORPGs are all based on hunting and gathering [http://mxac.com.au/drt/ProblemSolving.htm]. Any story is peripheral to the desire to gather gold, to get items, to gather more gold. In modern MMORPGs, this "gathering" is usually done by completing quests for people, who then pay you in gold, but even those quests often involve gathering something for the quest giver.
Right now, the story only advances because each new quest giver is in a different place and has a different story to tell. It's like seeing cinematics between each level, only these cinematics are boring, skippable, and have no effect on gameplay. Compare this to Fallout 3, where the only reason you are doing anything is to progress the story (which, consequently, increases your XP and makes you more powerful), and without following the story, you won't be able to make reasonable progress. In order for an MMORPG to integrate a story like this, it would need to change its motivation: leveling needs to be a result of participating in the story, not just doing odd jobs. Some of this may come down to writing better quests: most WoW quests have nothing to do with the story, but they could. Others may come down to placing less emphasis on hunting/gathering and more emphasis on the story. For example, maybe you couldn't advance to a given area until you finished a certain story-related quest (such as The Dark Portal in WoW -- one of the few places the story was really well integrated with the game).
In the vein of making the players part of the story, here's an idea. Just like in most MMORPGs, there are enemy encampments near town. But they don't just sit there: they attack at varying intervals and with varying numbers. If they manage to take the town, the entire landscape changes: you can't buy things from the stores and all the quest givers flee and won't give you any other quest than "take back our homes!". All of the players would have to band together to take back the town, and they would all want to do so because it's the only way they can continue playing the game. I imagine this might be difficult to do well and might even break the game ("Halla is taken? You mean I can't play the game until 20 people ome and help me? Then I'm not playing."), but it would certainly tie together the social/multiplayer aspect and the story line. Basically, make the NPCs, and the story that unfolds through them, depend on the state of the world, and make the state of the world dependent on what the players are doing. Halla in World of Warcraft does this well, but without fitting itself into the story.
Actually, there is one MMO that does something like this: Lord of the Rings Online. Sure, you end up spending a lot of time doing silly fetch quests, but there are also great moments where you are part of a huge battle and the entire world around you changes as a result. It requires instancing, which breaks the multiplayer experience a bit, but it's at least an attempt to accomplish this. Now if only the other 99% of the quests were as tightly integrated.
