I run mbsync/isync to keep a maildir copy of my email (hosted by someone else).

You can run it periodically with cron or systemd timers, it connects to an IMAP server, downloads all emails to a directory (in maildir format) for backup. You can also use this to migrate to another IMAP server.

If the webmail sucks, I wouldn’t run my own. I would consider using Thunderbird. It is a desktop/Android application. It syncs mail to your desktop/phone, so most of the time, it’s working with local storage so it’s much faster than most webmails.