An AMD E-350 based build might be worth some consideration, it has ultra low power consumption so that's good with it being on all the time and also helps to make small and quiet PC's. The performance isn't exactly fantastic but it is supposedly enough to play any kind of 1080P video.
In all honestly, I have one in my laptop and I just tried it. It seems to struggle but I haven't tweaked any settings or looked at it in any detail yet. I googled it and everyone seems pretty sure it's perfectly capable.
That only comes to about $250 and it should be cheap, quiet, have ultra low power consumption, it's small and has a pretty decent amount of storage for the price. The performance isn't exactly going to blow your mind though.
As long as your player supports hardware acceleration, most modern cpu's and onboard video should be able to handle 1080p video.
My media center while not ideal seems to work well, but is not silent(damn power supply, fan swap helped lots).
gtx 650 ti(cyclone II cooler. much more quiet now)
SG05 case with some extra mods. I do not think this case meets your small as can be needs.
128gigabyte M4 SSD Boot
WDC Blue 1TB media(not as quiet as my old Seagate)
I would for sure look at the AMD APU's as they offer better video without needing an extra card(honestly 1080p is not that hard any more). This allows smaller cases. Just ensure you can get a decent cooler to fit because the AMD stock coolers tend to run loud(if the system is loaded).