Helping people with computers... one answer at a time.
Backing up a free email account is best done by downloading it to a PC-based mail program. Some services support it, others need a trick or two.
Yet another reason I intensely dislike relying on free email accounts for "important" things. There are so many things that can, and do, go wrong it's scary. I hear very sad "I've lost everything" stories on a regular basis.
So you're quite right to want to backup.
I'd be remiss if I didn't start by recommending my original article Are free email services worth it?. If you're using a free email service as your primary email account, or even worse, as your only business account, you seriously need to reconsider what you're doing.
A pop3 account is, by far, your best way to backup your email. By downloading to your own machine, and then backing up the downloaded mail, you have everything under your control.
But what if, for whatever reason, that's not an option?
Well, your options are very limited. Basically you can try to automatically forward all email to another email address, or you can try to download your email. By forwarding your email to another account on a different free service, you basically got everything backed up at that service. Similarly, by using pop3 access to your free account, you can treat it like any pop3 account and download and back up your mail yourself.
Unfortunately from what I've seen, none of the free email services support either approach directly.
With one exception.
Google's GMail allows you to automatically forward incoming email to another email address, and it allows you POP3 access as well - meaning you can use a traditional mail client on your PC to download and read your email.
So if you're a GMail user, you could create a HotMail account and automatically forward all your mail to it. Or you could configure your PC's email client, like Outlook Express, and periodically download your email to your PC.
But what about other services like Hotmail, or Yahoo? How do you backup that email?
The best I've been able to find so far is that there are services on the internet that claim to be able to forward Hotmail and Yahoo mail, and services and tools that can provide POP3 access. I've not used any, so don't have any specific recommendations. Some are not free, and if you're going to spend money, I'd rather have you get a "real", supported, POP3 email account. I also believe that in most cases in order to use those tools, you'll need to provide your account information; something that, quite honestly makes me nervous. And in all cases, these approaches are not officially supported by Yahoo or Hotmail.