This may be too late for gandrusz but, for reference...
The answer depends on what you mean by "very small power requirements".
I think the easiest and safest solution would be to get a D510 or D525 based barebone such as the Shuttle XS35. They're silent (until you put in a HD), fairly cheap and reasonably powerful. They're regular x86/x86-64 processors and they're fully supported by most variants of Linux (there's stuff in the kernel that targets them). The downside is that the power consumption isn't that low.
You can of course build yourself something with a different case or just nail the board to a piece of furniture. If you're going to build your own, be aware that there are similar boards which have integrated power supplies.
The D945GSEJT single-core Atom board consumes significantly less power (see SPCR's review) but it's 32-bit only and I don't know if it has GbE. Or you could go with a D410/D425 (the single-core versions of the current desktop Atoms) if you wanted to save a bit of power. I don't know if anyone's offering the mobile versions of the current Atom line-up on a practical and affordable board. This might be a plan.
An advantage of this solution is that it should be easy to sell the gear or to transform it into a gift since these double nicely as word-processing/web-browsing desktops (especially the dual-cores).
If you're willing to take chances with dicey Linux support, there are devices such as the fit-PC out there built on the low-power Atom platform (the CPUs have Z* designations). These are 32-bit only and I don't know if they have GbE. The fit-PC are barebones which support a 2.5'' HD and they consume way less than a regular Atom system. Perhaps there's a detailed report about Linux support for these out there.
Then there are ARM devices and boards. I don't know any built device which supports an internal 2.5'' HD (do you?) unless you count the 3.5'' NAS enclosures which may waste more power than necessary. Power consumption should be really low but you may waste quite a bit of time to get one of these working and you may not be able to run your fave distro (or even any regular distro) on it. But if you want something really simple you're not going to be touching often and if you're willing to forego the hard drive, one of these might make sense.
Here's someone experience (found on wikipedia): http://www.earth.org.uk/note-on-SheevaPlug-setup.html
Hopefully someone will make proper ARM servers available someday. There are plans which are being talked about but I'm not holding my breath...
There are other offerings on the market based on VIA processors, ultra low-power AMDs and so on but I don't know how much power they'd likely consume and whether they're any good.