Sure, you can, but that's an entirely different question than if you should. These kinds of sites often don't last long on shared servers, and can very easily result in your site being disabled, your account being suspended, or terminated for TOS violations. The problem is often not the bandwidth, or the disk storage, but one or more of the following:
1) DMCA problems - users uploading copyrighted content and/or media without the distribution rights (music, warez, movies, images, etc.)
2) Heavy hammering of the server by multiple simultaneous HTTP requests (concurrent connections) rendering the server unusable for other users.
3) The presence of other illegal (in the United States, or the State of California) content (child porn, beastiality, etc.)
In short, what you are basically doing is opening up your account for use by others for whose actions you will be held responsible.
You should check out the DreamHost Abuse Center, as well as the TOS, for a better indications of the types of issues you are likely to have to deal with.
That said, depending upon a combination of features such as the sophistication of your software, access restrictions, regular and thorough policing of activity, and traffic management you may do just fine (though you will almost certainly be a candidate for a dedicated server if the site is at all successful).
Many others have found it too problematic to manage properly, and there are several "horror stories" on the web about users not understanding this and feeling angry and betrayed when their sites were taken down and/or accounts suspended or closed because they failed to recognize the impact of the problems I have mentioned and the realities of being on a shared server, but YMMV.
IIRC, the last fellow that offered a "general download site" and discussed it on the forums lasted less than a week as a result of TOS issues (and he didn't even allow public uploads!).