10Tb = 10,485,760MB. Let's assume that you're going to do this over a period of a month, thats 341GB a day, 14GB an hour, 242MB a minute, or 4MB a second. This equates to a 32Mb connection, fully maxed, 24 hours a day, 7 days a week.
Good luck getting that kind of service on a standard highly paid web host, let alone a $10/month host. They would kick you off after 24 hours I'm guessing, if they even had 32Mb of sustained bandwidth avaliable to a single site.
For that kind of traffic, I think you're best looking at a co-lo facility and your own 100Mb link. This is, of course, going to cost muchos dollar (lots of money), and might not be suitable for a free service.
-- Update --
Just saw the 2nd part of your question regarding streaming. We offer a streaming service for our clients, and they have a few hundred hours of lectures that they offer. We have about 100Gb of FLV files served by Flash Media Server. FMS is a reasonably cheap piece of software to license, it's easy to use, and there are plenty of free Flash-based players that will stream content from it immediately.
You can get Content Service Providers who will host and stream your videos for you. I believe you pay for disk space AND bandwith (as you would with a normal host), but it's fairly pricy. However it gives you a much more professional and redundant backbone.
We didn't bother, we just stuck it on a normal server running Server 2003 and stream it from there. It needs a lot of RAM and you probably won't get away with it on a shared host. You might get a VPS that you can configure yourself (FMS is avaliable for Windows and Linux hosts), which would also work fine.