How much traffic and usage will your site require? In most cases it is best to go for a VPS depending on the amount of traffic you'll receive. Some providers don't allow file sharing services to ran on their network so be sure to read their AUP and other terms before signing up.
I suggest you to get a cheap VPS for file sharing website.
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Shared hosting will not do good for you. You'll end up getting your account banned for using too much resources on shared hosting and usually most hosts won't even allow you to do file sharing hosting on shared servers.
Vps would be your best option here any small unmanaged vps will run you around $5 - $15 a month. If you don't know how to manage a server you can then go for a managed service which will cost around $20 a month or so.
I think even a basic vps plan would suit you. Don't go for shared hostig as your business would be not be allowed by most of the providers and in case if it is allowed there would be many limitations. Under vps you won't have any such but at the same time make sure you don't host any illegal content.
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You may be able to get away in the beginning with shared hosting. But I'd recommend just getting a vps. If budget is an issue for a vps take a look @ leaseweb, intovps, or vps6. Those three provider's are too pricy and they offer a good bang for your buck.
Just check their terms and conditions. I'd suggest hitting ctrl + f if your on chrome and scimming through their t&c to see if they deny a certain type of site, if they don't you have the right to do it!