Can I bring a shoebox PC in my carry-on bag?

I frequently fly all over Europe with shoe-box PC’s or various measuring devices in carry on luggage.
The measuring equipment is very expensive: € 250.000 is pretty normal for a single device. You don’t want to let it out of your sight or risk a baggage-handler to throw a suitcase around.
And they don’t like temperature fluctuations much (calibration issues). Having the suitcase sit on the tarmac for half an hour at -10 Celsius, while waiting for the plane to be loaded, is not a good idea.

I never had any serious problems. Being polite and patient towards the security people helps. And make sure you are an hour or so early for your flight so you have time for some delays at the security check.
If you are in a hurry to catch a flight, you probably get nervous or agitated at the security checkpoint. Security people will notice and will threat you with extreme suspicion by default. They can’t help it. That’s the way they are trained.

I always put my special stuff in it’s own tray through the scanner. Don’t stack multiple items on top of each other.
Sometimes I’m asked for details what I’m carrying or I’m asked to turn it on to show it is a working device.
And 4 or 5 times I was asked to open the (metal) case of the PC because the scanner couldn’t penetrate. (Save yourself some hassle and replace case-screws by thumb-screws so you can undo them by hand. You can’t carry a screw-driver in carry-on!)
Some devices (barcode scanners, FLIR scanners) look a lot like taser-guns, but nobody ever remarked on that.
If someone starts moaning about batteries just point at all the laptops, phones and tablets other people are bringing along.

I have no idea if this is standard practice everywhere or just anecdotal.

I had a special issue in 2011 at Amsterdam Airport:
A very sensitive radio-signal analyzer that couldn’t be X-rayed because that would cause it to loose calibration. Re-calibrating the thing would take 3 days and cost something like €50.000.
I had documentation from the manufacturer with me, even a hot-line-number that could be called for clarification.
When in front of the check-point I didn’t queue, but went straight to the security guy standing aside keeping an eye on the entire check-point. (There is always at least one…)
I explained the situation. He called in a supervisor. The supervisor directed me to the police station at the airport.
The police manually checked my carry-on luggage (very thoroughly) and called the manufacturer hot-line, but they used their own phone-number list, not the number I supplied. (I asked later and was told they had a list of most major manufacturers of medical equipment and measuring devices. Apparently this sort of thing is not that uncommon at large airports.)
They checked the device model and serial-number with the manufacturer and confirmed it was indeed that sensitive to X-rays. The manufacturer also confirmed the device was registered to my employer.
They filled in some paperwork detailing the device, serial-number, short description what it looked like and that it had been cleared by them. Some official stamps and signatures.
From there I went back to the checkpoint and handed the device and the paperwork to a guy at the X-Ray scanner. He took it to the other side and I followed the normal routine with the rest of my carry-on luggage. On the other side I was handed back the device and the paperwork. No problem.

I didn’t fly back from Milan with the device. I needed to go to Munich for another job. My colleague took the device back to Amsterdam. He followed almost the same procedure at the Malpensa airport:
He went to the airport police station straight away and explained the situation. (He had to wait 20 minutes before they found a guy with a decent grasp on English…) He also got the manual check, but here they didn’t bother with the manufacturer check. (He did show them the paperwork I received at Schiphol airport, that might have helped.)

I’ve carried a Transtec Senyo mini PC (http://www.pcpro.co.uk/transtec/transtec-senyo-610/29152/transtec-senyo-610-review) to and from Iceland many times. Usually with a laptop and an eee-pc in the same bag. I just make sure I take out each device and put them in trays just making sure that they aren’t on top of each other (although I’ve never had issues with more than 1 in a tub). I’ve also taken home made electronics in the same bag and not had an issue.

To put the amount of stuff into perspective between 2 of us we needed 8 large tubs to put through the x-ray machine, and 4 of us going through the same queue used all the tubs available…

In general it is not forbidden (I suppose there could be exceptions, comment if you know any!), but you may get a bit of scrutiny. You will not be the first person that day to have large electronics on you; IT contractors bring all sorts of electronic hardware through security checks. I suggest taking it out of your bag and putting it in a separate tray, so that it is easier to look at on the xray monitor and you are less likely to be delayed.

I’ve brought computer parts resembling what you have onto the plan before without issue. They didn’t even take a second look at it. You should have no problem, though the one part of the PC they might have concern with is the UPS.