I figured out what was going wrong when I tried to setup my account on Xbox Live and the solution was ridiculously stupid! This was my problem:
I created a Windows Live ID using my PC and responded to the confirmation email. [Fungii000]
I tried to use this ID to connect to Live with my XBox360. It asks me for an email and password. But when I submit this information, the exact error I get is "Xbox Live signup is currently unavailable. Please try again later".
What was the problem? The virtual keyboard you use to type in this information has what appears to be a "." key at the bottom, in the same area where the period is on a regular keyboard. This is not a period, but some other character, an underscore most likely. Although it's hard to tell, even at 1080p resolution! I entered my email address again, being sure to use the period key from the virtual number pad this time, and it went through no problem. Doh!
Too bad MS didn't see fit to give a more pertinent error message rather than the completely inaccurate, not to mention misleading "Xbox Live signup is currently unavailable. Please try again later." Sometimes I think they do this shit on purpose just to fuck with us. Bastards!