Would like to do this. Not just streaming but also with OTA TV as well. If I put an antenna on my roof I can get substantial amount of stations.
It is SO dang easy... Just find an old computer - A med-to-high-end dual core is plenty for power... That puts you in late WinVista to early Win7 machines (or better of course)... I went for a mini-tower for ascetic reasons, but really, anything will do... BTW, I AM going to a mid-tower - Since the media server is always on, it is a good back-up device, so I am going to be backing up my main server onto the media server (and visa-versa)... I'll have a nice cabinet for it to live in by then, but I digress...
Optimal bang-for-buck would be
a win7/64 based system (Win32 can't utilize multi-core processors properly). If the machine you harvest is not a 64bit system, it isn't a deal killer... Mine ran in Vista32 for almost two years. But I did run into a set of numbers for win7/64 recently (form another machine I junked), and it is well worth the reload.
an HDMI-out - Obviously necessary to connect to a modern TV - If you don't have a flat screen with hdmi ports by now, You can still do this (tougher, but ask how), but most folks do... If the machine you harvest doesn't have hdmi out, still not a deal killer, but you'll have to buy a vid card in order for things to work. Note that a large flat screen would benefit from a vid card upgrade anyway... Std onboard vid will run well on somewhere close to a 42/48" TV... Above that, a modest card (say around a 2g [around 30 bucks for a decent re-branded card]) will probably be needed. If you're a gamer, you already have a bomber card, so not to worry.
a Blu-ray DVD - throw out your DVD player and let the pooter do it. One of the BEST things I found out about this jump was the elimination of most all of my set-top stuff. I now have ONE clicker, and ONE keyboard... If the machine you harvested doesn't have blue-ray, still not a deal killer - If you don't do movie rentals, or play PC games, it doesn't matter at all, but otherwise, you'd have to buy the DVD component - Maybe $45.
a Wireless NIC - If you have a LAN wire close to your TV, disregard this (wire is always better than air). Most boxen, especially the cheap used machines available, are not going to have wireless onboard. You'll need it to get to your router and the internet. I bought a bunch of crappy lot-end 'N' series usb2 wireless devices for literally nothing (I think I am in them under 5 bucks) so I really don't know what to tell you for cost... maybe 15-25...
a Wireless Keyboard w/ glidepad - Wallyworld has a Logitech for $25. Not an absolute necessity, but if you plan to veg in your chair and surf, obviously a must. Another option that I also use is a VNC client, where you can control the TV box from a laptop or tablet. It's almost necessary too - A lot of times you can't see the lettering on the TV from across the room, or maybe you want to fire a movie from your lib... Any real navigation is going to be easier with a VNC connection instead of the keyboard... But for normal navigation inside a browser (90%) the keyboard is awesome.
A big hard drive - An option, to be sure, but if you have a video or audio library, you'll want to have the room... This box WILL wind up being your media server. ..
That's really about it. Other than that, you'll have to find the sites that carry TV series and movies, but I guarantee you'll be able to watch just about anything. The only thing I can't find so far are NASCAR events (which you can pay for to access) and One News Network (which you can pay for to access).
You might like a digital antenna too - It would be nice to get at local news... I haven't gone down that road yet, but even so, I can tell you that you'll need a good mid-range to high end antenna for decent success. Here I will only get 8 channels anyway, but in the city, I hear there are a lot more.
And lastly, I'd like to say that I kept my cableTV for about 3 months after I had my media server functioning and in good trim... Your TV can do both (just switch HDMI inputs), and it is nice to ease the transition. But after the first month (wherein I made myself do streaming), I soon found I was using the cable less and less, to the point that I was happy to cut the cord by the time I did.