As standard I offer a 2GB email space on all new clients on their emails.
If he really wants to store all his emails they he must use POP3 and download them on his email client, from here he can backup via Dropbox is something similar. You might need to configure your email to delete downloaded emails, however, by default once you've hit your limit it will automatically delete emails, so their online storage would be cleared up, however, all downloaded emails will be present on their email client (presuming it's configured with POP3).
I had a similar issue with a client. I resolved it by setting up their emails to Thunderbird and downloading a backup plug-in on their Thunderbird. I just let them know they would need to backup their files prior to formatting their system if they must, and they can easily put them back, keeping every email they ever downloaded.
I further explain that webmail will only show up emails on their server, and all downloaded emails will be shown on their Thunderbird. This is the only way to have 'unlimited space' as they would call it, which would be restricted to their computer's hard drive space.
If they wish to access their thunderbird from anywhere, you can configure it to run off Dropbox or similar service, but I've never done this as my clients have yet to demand something like this. For now they use a mixture of Dropbox and Thunderbird, either with POP3 to store all emails or SMTP if they wish to have messages synchronized on all devices.
In terms of your monthly fee, you cover many things, hosting, support, updates maintenance and domain charges.
Our hosting is mostly profit, but without it we can't justify the time we spend supporting, Just to highlight, I spend roughly 2 hours per day answering emails. If I did not do this I would have unhappy customers. I use a reseller account so I sell the hosting for x amounts and then take this as profit to support my clients.
Those clients don't mind. If they decided to go it alone they would still pay the same amount for a likely to be worse product. The time they will spend supporting themselves to the money they would save (no money) would not be worth the hassle. Let's not forget they would not get any updates on their plug-ins or technical assistance. The only support they would have would be from a hosting perspective, and in the end clients always demand support on their websites.
Don't let the client bully you either. 2GB web space is not that much, however, I would not recommend giving him any more. He should use a backup plug-in to store his emails on his local machine.
I really hopes this helps answer your question.