If you're using a shared hosting service, I recommend avoiding using Moodle's file management system as far as possible. It copies used files into /moodledata/ and serves them via a proxy script that can quickly fill up your server's memory limits and effectively block the site for other users. You may also have your site shut down by your hosting provider without notice (check their terms of service).
Also, most hosting providers don't like video sharing because it can quickly fill up terrabytes of storage space. Shared hosting providers usually set "unlimited" at 10-20GB.
Off the top of my head, I'd say that a couple of options would be:
#1 - Use a bucket server, which is kind of what you're doing with DropBox. There are others that may be more suitable for video storage and playback, maybe Amazon Web Services' S3? I think Moodle has some integration with that. Check out the repositories section in your Moodle and in Moodle.org plugins.
#2 - Use a media server service: there are some good integrations with Moodle, although this can get expensive.
#3 - You can make Youtube.com videos "unlisted", i.e. they don't show up in search results or on regular Youtube.com pages. Users need to know the exact URL to view them. Youtube.com has some fairly robust and easy to use upload tools too.
#3.5 - Vimeo.com (no ads on your videos, yay!) is Youtube.com's more creative and serious competitor and they offer paid for hosting options with privacy controls.
Can anyone think of any more?