If your iCloud mailbox is full
If you use the Mail app on your Mac to receive iCloud Mail and your iCloud storage fills up, you won’t be able to receive new email until you make more iCloud storage space available. You can do any of the following:
Delete iCloud Mail messages. See the Mail section of the Apple Support article Manage your iCloud storage.
The Mail app on your Mac stores email in iCloud because, by default, Mail uses IMAP for your iCloud email account.
See the Mail User Guide to use Mail on your Mac.