This actually seems to be a legitimate use of Emacs as pid 1!
It’s been done, but I don’t know if you can synchronize your files.
Just write an Elisp client for Dropbox, and done.
Sadly I don’t think the same technique to compile Emacs statically has worked since version 22 onward.
I use an iPad 4 mini with an microsoft universal keyboard for that.
I’ve got a scribble about that here: http://yakshav.es/mobile-working-setup/
There is also Hemingwrite, intended for writers.