Based on our experience, the following free IMAP-enabled email services have proven to be great destinations for email backups.
You can either create a dedicated account on more (or more) of these services to use for mail backups from your other accounts, or you can create an account that you use, but also store backups on. We're pretty sure that the email providers prefer the latter case. NOTE: using any of these services solely for mail backups may violate the service's TOS; please verify before using.
