Newegg has a couple of Lian-Li Mini-ITX cases with many drive bays like the
$120 Lian-Li PC-Q08 (reviewed
here), but I agree that Micro-ATX cases would be cheaper and more plentiful.
There are some interesting Mini-ITX motherboard+CPU options.
The
$125 Asus E35M1-I has six SATA III ports with the appealing
AMD Zacate E-350 APU.
The
$187 Jetway JNF99FL-525-LF has six SATA II ports with the Intel Atom D525 CPU.
The
$140 Zotac H67ITX-C-E has two SATA III ports and four SATA II ports. Its LGA1155 socket would accept a
$57 Celeron G530 or other Sandy Bridge processor.
Nearly-universal
1x4 GiB for $20 or
2x2 GiB PC3-10600 for $24 or
2x4 GiB PC3-12800 for $40 (CAS 9, 1.5V) 240-pin DDR3 desktop memory should work with the Zacate and Sandy Bridge options, but the Atom board would require
2x2 GiB of 204-pin DDR3 laptop memory.
Even a small energy efficient ATX power supply like the
$48½ SeaSonic SS-300ET is likely to be overkill for this low-power application.
In micro-ATX, there are many possibilities. A micro-ATX tower case like the
$118 SilverStone Temjin TJ-08E (reviewed
here) or a flimsy
$35 case could do the trick.
Zacate still offers an appealing low-power solution like the
$127½ Asus E35M1-M Pro with included E-350 APU.
You could consider Llano in socket-FM1 like the
$120 Asus F1A75M Pro with a
$50 A4-3400 or
$45 A4-3300.
You could consider socket-AM3+ like the
$112½ Asus M5A88-M with a
$65 Athlon II X2 260.
You could consider Sandy Bridge in LGA1155 like the
$130 Asus P8Z68-M Pro with a
$57 Celeron G530.
If you don't already have some old parts lying around to re-use, either Zacate or Sandy Bridge would be the most appealing option. Atom is slightly slower and much more expensive than Zacate while using slightly more power, so there's no good reason to consider Atom for any applications anywhere. Socket-AM3 processors are more power-hungry than Sandy Bridge. Socket-FM1 APUs use less power than Socket-AM3 CPUs, but you don't really need the graphics performance that Llano provides.