There is a major bug that is causing this problem to occur. It's due to some bad programming either by Square-Enix or Microsoft (they both blame each other).
The only way to resolve this issue currently is to perform a "Clear System Cache" on your Xbox Hard Disk. Please be advised that doing so will completely erase PlayOnline and Final Fantasy XI. After you have done so, re-install everything again. You will be able to play as you would normally.
This error only seems to pop up sometimes after you exit the game. A few members have said shutting off the Xbox 360 by holding down the power button on the Xbox instead of clicking "Shutdown or Log off" helps prevent this. But there is also a chance it will corrupt your data.
Basically this is a huge mess and hopefully it gets resolved very soon.