Hey there! Sorry to hear you’re having issues with the Mail app, maybe I can help!
How many accounts do you have set up? And if multiple, or even one, does it give an error message when attempting to send an email message to yourself?
I’d say if you hit Window > Connection Doctor and it says that they are all Online, there’s something else going on.
If it’s an authentication issue AND only happening with one account, I’d follow the steps JRU2 laid out. If it’s an IMAP account, the mail should be stored on the mail providers server, if it’s a POP account, even if you do have a Time Machine backup Is still export the mailboxes before deleting and re adding:
https://support.apple.com/guide/mail/import-or-export-mailboxes-mlhlp1030/mac
As with POP mail, this would physically delete the emails from the account, exporting them should save a local copy of them if something were to happen and they disappeared, IMAP is much more reliable.
Anyhow, if you can post the error and answers to those questions, maybe I can help. This can be related to the network, third party software, a corrupted keychain, permissions issues or even adware possibly (unintentionally) installed, (Or legit networking/security software installed).
But yes, most of the time removing/re adding the accounts fixes it, if not just let us know, cheers.