Email Backups:
Microsoft:
Outlook has a backup program for your .pst file. You can save a copy on an external drive as backup. This method allows a complete record of correspondence to be store somewhere other than you work machine.
Mac:
Mac has a commercial application to backup your emails automatically.
Alternative Backups:
You could open a free Google account and send a BCC to that account as a record.
WebMail is not recommended instead of a computer email program as that method tends to clog your hosting space.