How do I fix the iCloud "You can't sign in a this time" error?

I noticed that I wasn't able to view my iCloud usage on my MacBook Pro. To attempt to fix the problem I signed out of iCloud, but now I'm no longer able to sign back in – it displays You can't sign in at this time. Try signing in again. Additionally, I haven't been able to install anything from the App Store for about a day as it doesn't ask me for my Apple ID and password, and perpetually displays the throbber:

Reading lists and iCloud tabs are not syncing between my Mac and iPhone 6s either. Strangely, Messages and FaceTime are still signed in and working correctly.

So far, I've attempted to delete my iCloud password from Keychain, and delete the ~/Library/Application Support/iCloud/Accounts folder – both with no success.

Furthermore, I wanted to create a new Apple ID to test a fresh user, but the Create Apple ID link is greyed out and unclickable:

To sum up:

1. I can't log into my iCloud account.
2. Reading lists and iCloud tabs are not syncing.
3. I can't download anything from the App Store.
4. I'm not able to create a new Apple ID from my Mac.
5. Messages and FaceTime are not affected, and are still signed in.
6. I've attempted to delete my iCloud keychain entry, and iCloud account folder, with no success.

What else can I try? I've installed no anti-virus software, and know that I haven't downloaded anything suspicious to cause a virus or malware issue.