Yes you can if it's just HTML no server side stuff.
But you need your own domain of course and a way to change the CNAME record
To point to Dropbox's IP.
Personally I would store all the images and user generated content in
Directories per user on Dropbox on and have the main HTML
on another hosting provider giving you the best of both.
A small disk vps can be had pretty cheap and keep
The standard stuff / HTML/ logos/common resources there but the user generated stuff on dropbox.
I have a Large VPS for my main web stuff but use HOSTM shared account to play with livecode server.
Very good support and would work well with the Dropbox resource sharing.
It would probably cost less than $20 a month for both with 1tb of storage.
It won't batter you with eye watering price increases like AWS.https://en.m.wikipedia.org/wiki/CNAME_recordhttp://www.dropboxwiki.com/tips-and-tricks/host-websites-with-dropbox