I've built five PCs based on the DIYPC HTPC case linked by Ken, and find it to be an excellent choice IF you're not going any higher than an R270 or GTX960. You *might* be able to swing a 970 but airflow around the video card is limited. A blower-style cooler might allow a higher end card, but they're uncommon.
My HD7850, which is rated at 130w, runs about 5-7c higher with the side of the case on than with it off under extended torture with Prime95 + Furmark, with the rest of my fans tuned for silence. Not quiet, mind you, but silence. I'm pretty happy with it.
The case will support two 2.5" drives and a 3.5" drive, but I strongly recommend pulling out the 3.5" bay (above the video card) if you're going above a 100w discrete card without a blower-style cooler, as it aids airflow in the area. You could do a 2.5" storage drive alongside an SSD, or go with external mass storage.
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Off the top of my head, this is roughly what I'd recommend for a generic ITX build:
1) Haswell Core i3 4160, or perhaps a lower-end i5. An i3 is an excellent choice for emulation, and is undoubtedly appropriate to pair with a video card up to perhaps a GTX960 or R280. An i5 would be a fine choice too, depending on your use-case.
2) ASrock H97M-ITX - cheap as dirt, reliable, excellent fan control options, built in WiFi, all of the connectivity you could want. Like all other ITX boards it has only 2 RAM slots. I have zero complaints about these that I've used in builds.
3) 1x 8GB DDR3 low voltage RAM - Because ITX boards only have two RAM slots, you might consider a single 8GB stick of RAM to leave upgrade options. Dual channel doesn't make a ton of difference on an i3 except for a few specific cases. Low voltage RAM is a plus - go for a 1.25v or 1.35v kit to save a few watts.
4) Power supply - choose carefully. A quality 350w unit is plenty enough for an i3 + GTX970 without a ton of accessories. 500w will be massive overkill for any build this size. Gold rated units will be quieter and dump less heat. Some PSUs are shorter than ATX spec, and this will help with wiring. Modular is a plus, but not a necessity. Pick something with a quiet fan. If you're going with the DIYPC case, a PSU with a single 80mm may be a superior choice than one with a 120mm fan, because of the airflow dynamics in the case - it will suck air away from the video card, rather than from around the CPU. I've put Antec Earthwatts 380w PSUs in most of my builds lately as they changed the fan they use (it's a quiet, quality NMB unit now) and they're really cheap, often ~$25 after rebate.
5) The DIYPC case is great if you're content with R270 or GTX960 performance. The build quality is pretty good, and it looks great. Go with something with a horizontal motherboard layout and vents near the video card if you want to go with something faster. The case fan included has no brand on it and it's rather noisy even when undervolted, so I opted for a Noctua as a replacement. There are other cheaper fans of comparable quality, but the accessories included with the Noctua fan attracted me, and were useful.
6) If you're going with an i3, the stock Intel cooler will probably be fine. It has *excellent* acoustics at idle, better than most aftermarket replacements, though can get a bit noisy under an AVX load with an i5 or i7. A tower cooler will absolutely not work in most ITX cases. I opted for a Noctua low profile cooler, the NH-L9i. Although it's only rated for 65w CPUs or less, that's very conservative and it performs significantly better than the stock heatsink included with i5's and i7's, at lower noise levels. Another option is the Scythe Kozuti, which is priced about the same and has similar performance and acoustics.
EDIT: I'd like to add, I think you might underestimate how small something like the Brix really is. It will fit in the palm of your hand. Most NUCs don't even have room for a 2.5" laptop HDD and instead rely on M.2 form factor SSDs for storage.