A 'home server' may be suitable while your userbase is small, or if you are not depending on having it online all the time (power outages, maybe it is against the ISP's terms of service even?).
I guess it really depends also on how large the files are, and how many you have. I wouldn't recommend the home server because most importantly, the bandwidth speed and/or limit would be nothing in comparison to a real web host.
If you are looking to keep the web server to serve just pages, and you want a separate server for the media, there are a few hosts which are good for large files, and Amazon offers a very cheap and effective way to do this. http://aws.amazon.com/
For a live streaming server, you can look into a media distribution server which takes a stream from your own server and will mirror it to all available listeners, sparing you the bandwidth. [link]
I personally would recommend either looking into a static file host for the media files (like a CDN) or actually finding a web host that has a sufficient amount of storage and bandwidth, so you can serve the media files directly from the website.
