Q: Mac can't connect to iCloud because of a problem with [email]

Please sign out of iCloud in its preference pane, then sign back in. No data will be removed from the servers, and you can retrieve it by signing back in. That said, you should always have a current archive of the data for safety's sake, even if you don't sign out.

If you use iCloud Keychain, when you sign back in to iCloud follow one of the procedures described in this support article to set it up on an additional device.

Try going in System Presences > Internet Accounts > and on the iCloud account that you are using make sure there is the blue check mark on the box next to the one that says Mail. If it asks for your iCloud password enter it and it should work. Hopefully your issues resolves.

Do you have any other mail accounts, and if so, are they also affected? What about other Internet applications, such as web browsers?

I still don't know whether other Internet applications work when Mail isn't working.

And when that happens, can you load web pages in a browser? That's what I'm trying to find out.

Please contact Apple Support. When you set up a support call, select "Apple ID" as the product you need help with, not the hardware model. That way, if you're not under AppleCare, you may be able to talk your way out of being charged for the call.