Like all answers, it’s going to depend upon what you plan to do. There are a LOT of discussions like this on the openHAB Community site, along with specialist pages showing tested configurations.
That said, here’s what I run, what I’ll likely run next, and why:
Current config
Machine:
[ul][li]ODroid C1+ (1GB RAM, Quad Core 1.5GHz ARM, GigE)[/li]
[li]32GB eMMC Module[/li]
[li]Plus a “on-hand spare”, without the eMMC - I always have 2x of my critical HW :)[/li][/ul]
Services:
[ul][li]Vera 3/UI5 (Z-Wave, Paradox Alarm, Sonos)[/li]
[li]openHAB 1.8.x (MiOS Bridge, Prowl Notifications, Rules, MQTT Data acquisition/publication, 1000+ Items),[/li]
[li]Mosquitto (MQTT)[/li]
[li]emonCMS (Energy data & History with ~90 channels @90s intervals)[/li]
[li]DEBUG Logging perm enabled[/li][/ul]
Price: All in $110 (Case, Power Supply, eMMC Module, Taxes, Excluding the “on-hand spare” I keep)
Usage: 1% CPU typical, 20% at Peak (infrequent)
Why: No Fan, Low Power (<6W typical), Low Heat (it’s in a SW Enclosure, no fans), Compact, FAST IO
Why Not: How long will eMMC last?
openHAB 2.x/Eclipse SmartHome trial config
Machine:
[ul][li]RPi2 (1GB RAM, Quad Core 0.9GHz ARM, 100MbE)[/li]
[li]32GB SDXC Module[/li][/ul]
Price: All in $64 (Case, Power Supply, SDXC Card, Taxes)
Why: No Fan, Low Power, Low Heat, Compact. It’s largely disposable. I already had it. It runs quite fast
Why Not: SDXC Cards are SLLLLLLOOOOOW, and wear out quickly/easily.
Notes: I’m adding a 250G mSATA piggy back to this shortly, just for fun 
Original config
I ran this when starting off with openHAB, to get a feel for how well it worked, and if it was “for me” or not.
Machine:
[ul][li]VM Under my Mac Mini (1GB RAM, VM slice of a Quad Core i7, GigE)[/li]
[li]VM Slice of my SSD[/li][/ul]
Price: $FREE - I already had the machine, it’s already on 24x7 and uses ~24-30W on it’s idle period.
Why: Why spend $$ to trial something when you have a machine on hand
Why Not: My HA shouldn’t go down because I need to patch my MAC
Next config
Probably a NUC, but a tiny slice of it since I need it for other VM Testing (KVM, ESXi and Docker), along with consolidation of a Windows machine and a few other server pieces from around the house.
Why: They’re cheap, and power-scale IF you have a lot of things to consolidate.
Why Not: Typical user’s don’t need this much power, even the entry-level Celeron & Pentium models are OTT.
Notes: These machines are cheap, but they’re typically sold as bare bones config, requiring $$ for Memory & Storage (etc). The models I’d use (Gen 5/6 Core i5/i7, 16-32G RAM) are more geared to my other uses of it (VM & Hypervisor Testing etc) the baby ones would be fine for typical uses.