I don't think anything below a dedicated server (or a powerful virtual one) will do. And they do not come cheap. You could also host your own server, but there, again, you will need a fast internet connection (like fibre optic) and backups and redundancy and the like.
Note that you will not only require a server, but also support or a tech team to solve performance issues.
These are all things that you do not get with cheap shared hosting. I suggest you do more research on this matter. But my bet is that unless you invest some serious money, you do not get to host your own file sharing service.
If you are serious about Filehosting, you might want to look for filehosters that offer some kind of reselling program with branding (i.e. you can use your own label). A list of Filehosts can be found here: http://premium-account.org/en/1-home.
I was not able to find such an offer quickly, though.