I wrote up a big blog post about reverse-proxying to a Drobo FS (or Drobo Pro FS) using Nginx and lighttpd. You could disregard the reverse proxying portion and simply set up lighttpd on the Drobo as I describe, which will get you where you need to be. You’ll have to enable https on lighttpd yourself, but that’s quite easy to do with Ricardo’s lighttpd port. Or you can do what I did and use another web server with https already configured as a reverse proxy, which moves the (admittedly small) burden of doing the SSL/TLS encryption off of the drobo and onto the web server, and also doesn’t expose your Drobo directly to the Internet.
I also didn’t screw around with WebDAV at all, instead relying on lighttpd’s auto directory listing. Of course, that means I can’t upload files via the web interface, but I have no need of doing that.