Subsonic Forum

Is it possible to use Subsonic on a hosted server like Host Gator or online storage like Dropbox?

Not really. You would have to find a hosting provider that would let upload a JAR file and run it on their servers. Host Gator isn't designed to let you do that. Shared hosting plans are given very little CPU so even if you could your install would be unbearably slow. Their are some for hosting Subsonic 'in the cloud'
http://www.subsonic.org/pages/hosting.jsp

A better and cheaper option is to find a VPS provider you like (like digitalocean) host it yourself with a VPS.