I just got this error today, but in an odd way.
It was working fine 30m ago. I was downloading updated versions of some albums. One of the tracks of one of the albums had a digital glitch in the original 128k protected version (Lustmord, Heresy Pt. 1 2:26 in). This error does not happen on the CD. After listening to the updated track, the error is still there. I went to my history to report the error, sent the report, and continued on. The problem is that now I get the 5556 error. I have no idea why. The iTunes Store did ask me to log in for some reason when I went to access my log but I was already logged in. No matter, probably a security measure. However, after the report was sent I can no longer download songs from iCloud.
Error 5556, I'm guessing, is a network error on the server side whose error code string was not baked into iTunes which is why it says "unknown error". I've seen several reports that people's accounts just started working again later. I hope this is the case with me. I'll still report this error to Apple.
EDIT: And yes, I did the whole log out, etc. That didn't fix it, nor should I have to do that to make this work.