It's worth noting that anybody who offers you "unlimited" storage/bandwidth/whatever is probably lying. They'll have far more storage and bandwidth than you'll probably need, but you'll be sharing it with hundreds or thousands of other users. If one of them gets a sudden spike in traffic, it could hurt everyone's sites.
Yeah...that's just a limitation of the "pipe"....but some hosting providers have limits, so you can only store and transfer so much per day/month. I found a couple that said "unlimited", but then in the details that say it doesn't mean absolutely unlimited though they do cover about 99.5% of their user's needs, and it's only when dealing with multimedia files....

...and that's when I stopped reading and decided they would not be good for me.
I found a three that look pretty good, way inexpensive, and they do all the streaming media stuff.... justhost, hostmonster, and bluehost.
ipage and fatcow (it's pretty much the same company)....look really good, but they are the ones who say this in their details about "unlimited":
How much disk space and bandwidth will my plan include?
There are no set limits on the disk space or data transfer (bandwidth) that we provide in our hosting plans. We want you to have the resources you need to build a great online presence, and 99.95% of all customers will have more than enough disk space and bandwidth to meet their needs.
That being said, we do require all customers to be fully compliant with our Terms of Service and utilize disk space and bandwidth in the normal operation of a personal or small business website. While it is rare, we may need to put constraints on accounts that are using resources beyond what would be expected in the normal operation of a personal or small business website.
Again, the most common situations involve customers who use their accounts for storage of files—particularly multimedia files—that are not linked off of their websites. It is very rare for a customer who is managing a personal or small business website to exceed normal usage.
I'm sure you can still move a lot of files with them and not exceed their "limits"....but with that disclaimer, they can put a lid on media storage & transfers if you do too much.
I often put media on my current website simply to allow someone else to download it....but I don't necessarily put *links* to it on my website....and that's what they are saying, if you are moving data but not linking to it directly off your webpages..it can be viewed as a file-sharing.
Anyway....I'm going to go with one of the other three I mentioned.