Well, considering the specialty of emails and its difficulty of finding and restoring, the word "backup" shouldn't really be considered and used. How many times you have successfully found and retrieved emails from your backup copies? And how much work needs to be involved to keep a good and useful backup copies for your emails? Preferably and ideally, every single email should be backed up at the time when it arrives in your inbox, not sometime after because then it's harder for you to certain whether or not they are all up-to-date. And that's why the idea of archiving was introduced and widely used these days, both for your personal emails and working emails. Time saved for finding old emails and thinking/doing email backup is just huge. Deleting emails without worrying about losing them permanently is just sweet.
For personal user, simply sign up another free email account on Gmail, or Yahoo, or whichever has the huge storage capacity, and set up the forwarding in the current email account to forward all incoming emails to the new signed email account. Make sure to still keep the copy in the current email inbox though. Most of public email service providers all provide this such feature, except hotmail. Hotmail, the first widely-adopted web email service, seems quite out-dated these days, even after a major upgrade to live mail. It lacks so much useful features comparing to its major competitors like GMail and Yahoo Mail. Basically, if you are still a user of hotmail, you are limited to yourself only use Outlook family to back up your emails because hotmail doesn't offer the forwarding feature to its users. Or it's time for hotmail users to switch over.
For corporate users, if you are an IT pro who manages your own corp email server it's time to implement an email archiving solution. With an easy-to-use reliable email archiving system in place, you can have a more efficient light-weighted running email server, implement email retention policy office-wide fairly easily, and educate end users how to organize emails more efficiently without worrying about losing any emails, etc. Besides, it's not an expensive solution anymore.