The reasons I would be slow to have any large number of emails on a server include:
1) The server does not have easily the folder intelligence that you create on-the-fly with your mailboxes on your own puter. ie. Post-filtering.
2) Local searching will be much quicker, and super-quicker if indexed. And will normally automatically be integrated into your reply and address-book system. Even if the mail is on a USB you can work with it very easily. I know with Eudora you can set up a separate shortcut for the data in another mappable location like a USB.
3) You can run into trouble trouble-shooting the server downloads when the server is gagging-full. e.g There could be some arcane difficulty hanging the server with email number 5000. Very hard to troubleshoot.
If you want to have to have email archived, it is best after you have it backed up in the format you most like. These would include.
1) .mbx or .mbox or proprietary or MailStore .dat ... whatever condensed, sensible format you like
2) After everything is filtered
3) After anything you want deleted can be out first, you may want, eg. to eliminate thousands of attachments at once if your email program keeps them external to the text. Eudora eg. has two easily searchable folders, one for embedded pics, one for attachments. Or you may want to delete a lot of spam and junk mail. Again, easier to do at home than on the server.
Incidentally, Eudora is also fast with a gazillion emails, although likely not as techie-fast as TheBat! With Eudora I think we consider it good practice to keep the individual folders below 30K email.
Once you are done, if you want cloud backup just go ahead -- just as you would any other large files.
Now somebody has the server space free because it is bundled with a domain. So I could see how you might save a few dollars over other cloud backup. I think if you search around you can find cheapo just fine, and save the hassles above. And especially, backup locally.