I am planning a simple video sharing module on a project where a user can upload a video on our module using his Dropbox and is able to show it through our module. I believe I can do using their API and in fact I guess some mobile apps do use Dropbox for sharing/accessing user files. So, is it possible?
Yes, it is.
Setting folder access to publicly shared allows anyone to embed, access and view the content within, but keep in mind, Dropbox will throttle access if you link a 50MB video and you use an insane amount of bandwidth to stream it. The chunked_upload function may be of interest to you.