Jeff, unfortunately this looks like a problem with your hosting setup, so your host will be the only one who can help. If they can’t, and the problem doesn’t correct itself, I’d recommend exporting your site and moving to a more supportive host.
In the meantime, though, who is your current host? If you have FTP access, try manually updating a plug-in by transferring the new files to your server. If that works, then you can manually update WordPress using the standard upgrade instructions.
If you don’t have FTP access and still can’t auto-update, there’s not much more we can do without going through your host directly.