Sounds like you could get away with a pretty low spec box for the Apache Server side of things, any old Pentium-based box should do. OTOH, the MythTV box spec is a bit more interesting. On the basis that you probably are looking at a single box, rather than having a MythServer somewhere away from your TV and a small MythClient actually delivering media to the TV, you need a low-noise, low-heat box that looks like a PVR, not something out of a server room! This is a bit more interesting to spec!
I have done some work on MythTV, and have found that the Haupaugge TV cards work well. You can get good results with a mid-range video solution (I was, at one stage, running successfully with an ATi Radeon Xpress 200-based, on-board video solution). The difficult bits really are containing the noise, and ensuring there is sufficient cooling. There are some useful facts on
http://www.projecthtpc.co.uk/htpc_hardware.html which, although UK-oriented and running Windoze, shows some thought has gone into component selection, and should adapt to running MythTV (of whatever Distro) quite well, even to the point that there is a solution to get the VFD and volume Knob working. (
http://www.mythtvtalk.com/forum/arch...c_fusion_.html)
BTW, if you have the common-or-garden variety of Broadband, you will be running with a dynamic IP address, which will make accessing the server from the outside world a tad awkward (not to overstate the case, neraly impossible with reliability!), but there is a solution, have a look around DynDNS.com, that worked well for me till I went to static addresses at home.
Finally, and apologies if I am telling you what you already know...Use a firewall! If you are doing simple stuff, Firestarter is GUI-based and has a real gentle learning curve. If you want to do clever stuff, Shorewall does it for me, and its learning curve, while not as gentle as Firestarter, does not require rope and crampons. When you have it all set up, use Shields Up! (
https://www.grc.com/x/ne.dll?bh0bkyd2) to check that you have secured the system as you thought.
Hope this helps, good luck!