Are you asking because someone told you it could be? Dropbox can share a static, html website, not actually be a computing machine that could run software apps like an email server.
If your question comes from a certain idea about how you might possibly do that, by all means reply and I'll see if I can explain further.
It was just a subjective question. I thought Dropbox could just extend this functionality.
Or maybe 3rd party apps/services could felicitate this just the way e.g. http://calepin.co does for static site generation. Use Dropbox for storage of emails. And naturally you'll have a local email archive on all your connected devices.
So, seems there's nth like this as Dropbox doesn't let us to do the compounding, as you also pointed out.
If your question comes from a certain idea about how you might possibly do that, by all means reply and I'll see if I can explain further.