As the saying goes, there's more than one way to skin a cat--or perhaps that should be, there's more than one way to back up your inbox and email archive. If you use Gmail, you can download it all to your computer for free with Gmvault. If you're looking for a cloud-based backup software solution that also takes care of your calendar data and contacts, you could use Spanning Backup. But if you only want your emails backed up, and don't want to download them to your computer, then Dropmyemail ($10-$70 per year, 15-day free trial, pricing always in US Dollars) may be what you need. See all: PC Advisor software downloads.
Dropmyemail is web-based, so there's nothing to download. It can back up any POP or IMAP email account, so you are not limited to using it only with Gmail. However, since many people use Gmail and Google Apps, Dropmyemail makes Gmail backup extra-easy. You can log in using your existing Google account, so there's no need to open a new account and remember yet another password. Once you're logged in, it only takes a couple of clicks to grant Dropmyemail permission into your email archive.
This was one of the few confusing parts in using Dropmyemail: I got an error message that told me to click a button "to the right," but there was no button there. I ended up clicking the large, obvious green button in another part of the window that said "Sign in with Google," (even though I was already logged in) and that did the trick.
Once you let Dropmyemail into your inbox, there's not much left for you to do: A progress bar starts filling up as Dropmyemail downloads your archive. Since the process is happening remotely, you can close the window or even shut down your computer - backup will keep happening.
The Web interface for Dropmyemail is quite slick. It lets you browse your emails, read entire threads, and search. Its most compelling feature is the file manager, which lets you browse attachments without having to go through email threads. For example, you can see all of the images you've received over email in one place, which makes it much easier to track down a specific image without having to remember what the email subject for it was. This is a bit like what Lost Photos can do, but folded into a backup application.
Restoring email using Dropmyemail is a simple process: Just click the Restore button, confirm, and the emails will show up in your inbox again. The only thing missing is a way to restore individual messages: Restoring is an all-or-nothing proposition, so you cannot restore just a single message or a folder (label) of messages.
Dropmyemail tackles the often mundane task of backup with a simple interface and some innovative touches, like the File Browser. It is just one player in a crowded market, but its 15-day trial should help you decide if it's the right solution for you.