Alright, here goes. I'm not very familiar with media servers and such, so I don't know whether it's supposed to do video decoding or just act as a regular file server, but it should be capable of doing both. I'll do a breakdown of the parts, as well as explain some of my choices. Oh, and I bumped up the form factor to MicroATX if that's okay.
Motherboard / CPUASUS A78M-E +
AMD A4-6300 - $85
Since I couldn't find any Mini-ITX motherboards with 6 SATA ports I went for a MicroATX board instead. I also upgraded the CPU to an A4-6300 since I read that the A4-5000 had some issues with streaming high-bitrate 1080p video. The combo is only $10 more, but since you don't need a SATA controller card you'll actually save some money. You also have the advantage of being able to upgrade in the future if you want to stream 4k video.
RAMTeam Xtreem Dark Series 4x2GB - $60
I don't understand what you'd need 16 gigs for; most guides I've seen have recommended about 4, though I went with 8 just to be on the safe side.
Hard Drives3x WD Red 3TB - 3x $112
No objections here, I think.
CaseNanoxia NXDS4B MicroATX case - $80
Since I chose a MicroATX motherboard I had to get a MicroATX case as well. The Nanoxia was the cheapest case I could find that had 6 3.5" drive bays. It's got dust filters too, which is neat. If you can manage with a full ATX case you can go even cheaper.
Power SupplyRosewill CAPSTONE-450 - $60
It's way overkill, I know, but this was the cheapest decent power supply I could find that had 6 SATA connectors. Despite being higher-powered, it's still more efficient than the Antec at the same loads, and it's also got all-Japanese capacitors and a two year longer warranty which is something you want in a PSU that's supposed to run 24/7. If cable management is important to you there's also a
modular version for $5 more.
Full list: http://pcpartpicker.com/user/Chipskate/saved/HgnMnQ
Total: $622So, there you go. $80 cheaper (probably more with the SATA controller you'd need for the MiniITX build), better performance and probably more reliable. As I said, though, I don't know too much about media servers, so you should ask for a second opinion from someone who's more experienced than me.
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