I had this problem for awhile and I did list it some time ago... but wasn't sure of the actual problem. I'm now aware that the Xbox 360 I bought online wasn't updated regularly and now stuck with it. However, I also recall that this didn't keep me from signing in to Xbox Live for a few months until the May 2011 update. (The warranty sticker is still intact so I assume nobody modded it) Now when I try to sign in I'm asked to update and when I do it halts when "Applying Update" appears on the screen and gives me an error message. I checked it and it associates to the description of "Looking for required update"... but I have no way of getting it as far as I know.
Here's the version I'm stuck with: 2.0.12625.0 (January 19, 2011 according to Wikipedia... at your discretion)