I dont know if at all i make any sense, but this popped up in my mind. Can we use the 2gb free hosting of dropbox to put our django app over there and do some hacks to run our app?

No, it's not possible.

If you want to use dropbox as the hosting, you can put static pages, meaning just the html files with CSS and javascript because those files can be retrieved by doing a simple GET request.

Hosting a python code is not possible as it requires server side calculations & dropbox is just for saving files.

See the official wiki for the available hosts. https://code.djangoproject.com/wiki/DjangoFriendlyWebHosts