I actually JUST signed up for the forums to discuss this because I think it is an awesome idea. I was even playing Minecraft while listening to the SGU, so it was just a tiny bit meta, even.
A friend and I built a small server from old components that he runs at his house and I administrate remotely. It has never had more than 5 people on it at once, so I don't know exactly how to scale it to what Jay and Rebecca were talking about on the show, but having done this before, here are my two cents.
If you want to build a machine for this, it is going to have to be pretty beefy for this scale. My server is on an old Athalon XP 2400 single core with 3gb of RAM. It runs fine most of the time with five people, but has MAJOR issues when even one person is exploring previously unexplored areas; during that time the server has to generate new terrain and it makes it virtually unplayable for everybody else. How does it scale up to 25? 50? 100? I couldn't say, as there are no real guidelines (at least any that I have found) on the internet. I would say that the best bet is a modern processor with as many cores as you can afford, since the server application can actually use multiple cores at once. Minecraft forums are divided as to how much RAM actually influences server performance, but RAM is cheap enough that I'd say the more, the merrier.
Hard drives aren't that big an issue, as long as it is at least 7200 RPM and SATA, it should be fast enough. I'd recommend either a second drive or cloud storage somewhere, if not both, for daily backups in case the hard drive fails.
You'll need a fast, reliable internet connection to handle bandwidth. You'll also want to avoid any usage caps that many ISPs are instituting these days, such as AT&T's Uverse which will have a 150gb cap starting Monday. I don't know specifically how much data is sent to and from the server, but if 50 people connect per day all the data is going to add up faster than you realize.
Setting up your own server would therefore need an initial investment for hardware, ongoing investment for the internet connection and electricity, and the effort of admins and moderators.
If and when this gets off the ground, I would happily volunteer to help administrate and set up the server, since I have experience both setting up the Minecraft server application, as well as the server's operating system. I don't have the resources to host it myself (with money, hardware, or bandwidth), but once it is set up, it can be administrated remotely, and I would be happy to help. And I won't lie, like the nerd I am, I would have a lot of fun doing it, too.
EDIT: I would also like to add that I would be willing to host a coordinated server stress test some time on my hardware (Core 2 Quad core w/ 4gb of RAM.) In other words, get as many people as we can to cram into a temporary server I would host and see how well it runs on the hardware I've got, since everything I stated above is based on pure speculation. If anybody is interested in helping, all they would have to do is sign on at whatever time we decide on and play for a while. Then we could at least get a better of idea as to what hardware an SGU Minecraft server would need.