How do I fix Mac Mail not downloading messages and greyed-out System Settings?

Mac Mail won't download messages and Mail doesn't work at all. Even System Settings in Mail is greyed out, so I can't get into my accounts. The system tries to download messages, fails, and then advises me to make sure I have sufficient space in my Home folder. When I check storage, I have about 80 gigs of available space. Thanks for any ideas you might have.

[Re-Titled by Moderator]