I just got done chatting with support, told him I couldn't check my email on my phone because iCloud was unavailable. The guy told me that the "System Status" page didn't show any issues. I then told him I couldn't sign in on my computer either. A few minutes later he responded that he was learning of some current issues with iCloud, and that they are working on fixing them. He recommended checking again in an hour.
This started at 10:54am EST, as that was the last time my email updated.