I dont know if it is possibly on your GoDaddy Hosting Account, but im not sure, you do need Python: http://support.godaddy.com/help/article/7254/can-i-use-python-272-with-my-hosting-account but I read you cant compile modules (as in http://support.godaddy.com/groups/web-hosting/forum/topic/installing-and-using-python-packages/)
I have a setup where I have installed Dropbox on the server and use it like you describe.
I will leave this as reference for you or other users who want to have Dropbox for syncing parts of a website or for e.g. offloading backups and logs automatically (so the other way around). In my case I use it for transferring / syncing some data files to my server that I then process with my website so it always has current data.
- get hosting*) where you can run the required Python Version (this depends on the version of the python script you are using for CLI against Dropbox so I cant give the exact version) (but depending on the one you are using take it along as a requirement). For me: minimal version 2.4 of the C library, wget and Python 2.5 (no support for Python 3)
- then install Dropbox on that server: always via https://www.dropbox.com/install?os=lnx (tar -xczf dropbox.tar.gz) : this will become /.dropbox-dist , you can now run this as ~/.dropbox-dist/dropboxd & to run it as a deamon)
- To link it to your account : you probably have no browser on your server so the handiest is probably via a ssh tunnel: ssh -D 9999 username@ip to link it
- the folder /Dropbox is then created and it starts syncing
- via the CLI tool you can then exclude directories that you dont want to have synced, POSSIBLY you only want one specific folder to be synced, how to use? like this: dropbox.py exclude add "blabla"
More information and other links etc... : http://www.dropboxwiki.com/Text_Based_Linux_Install
I ment to use it only for those data files but occasionally it is pretty handy to get some information e.g. logs quickly to my local machine while I'm in a flow. So its really handy once in a while other than only for the data files.
Note that you should not put your most securest information in that Dropbox sync, there are probably multiple "points" where someone in the worst case could get hold of that information so only use it for information that basically is not of interest for anyone and more or less public information anyway.
*) I fired up an Amazon image for this which costs me about $50 a month (Linux/UNIX Standard Medium Instance, 3.75 GB memory, 2 EC2 units, 410GB)