Is there a way to limit how much storage users have to store their content? I have a video sharing site where members can upload and embed videos.

When members upload videos, is there a way to limit how much storage they have to store their content? Is there an existing module or one that can be modified in order to achieve this functionality?

It uses FFMPEG to convert the videos in the format that is required and I believe it is using the video module. The storage API sounds like it has promise.
With a support request and a little bit of work, Storage API could probably do this for you. In Storage API terms, I believe what you would do is create a "container" per user, and then use the sub-module "audit" to check how much space is being used. This number in turn could be used by the field widget you are using to limit uploads.