OK, I think we may be onto something guys.
First, get yourself a standard car tyre valve - $2.99 for two from Autobarn. Then drill a 14mm hole in your cube lid. The packet says 11.5mm hole but they are dreaming.
Next drill a 3mm hole in the valve stem, just up from where it flares at the base, but not too far up - otherwise you'll hit the bottom of the valve inside. Make sure you drill through the brass inner as well so that you have a clear hole all the way through to the centre. Then pull the valve through the hole in the lid:
Next get a few inches of rubber gas tubing, like the stuff you had on your bunsen burners at school. It's about 7mm ID. I got mine from Gameco in Preston, but you should be able to find it (or something similar) elsewhere. You might be able to find something in an auto parts supplier. It just needs to be about the right size and have good stretchiness, there is room for fine adjustment later. Then cut yourself a small piece about 15mm long, and slide it over the valve stem (a spot of dish liquid will help). It should be a nice tight fit. Find a washer and a nut to fit the thread, to keep the rubber tubing in place. I'm not exactly sure where my nut came from originally but there's a good chance it is from a bicycle tube, so if you are having trouble finding one try your nearest cycle repairer.
...and basically you are all set.
You will need to tweak it to get the backpressure just right - adjustments are made by changing the length of the tubing slightly. A longer length will be tighter and hold more pressure, and a shorter length will be a bit looser obviously. You might even be able to use a diagonal cut on the bottom end of the tubing to get some kind of adjustment from simply rotating it - but you'll only be able to do this while the dish liquid is relatively fresh, after a few minutes all that tight rubber gets pretty grippy.
From my experiments, I reckon about 3-6psi results in a firm cube with gentle bulging of the flat sides but no serious distortion. It should still sit flat on a surface. Any more than about 10psi will begin to seriously distort the cube, which agrees with what Shunty found - but the objective here is only to maintain a small amount of positive pressure to keep a base level of condition in the beer.
Using one of these it should be easy to naturally prime in the cube, with any excess pressure buildup venting safely. Then you just need to use a bike tyre inflator to give it an occasional squirt of CO2 to maintain positive pressure as you empty the cube (or if you're feeling really minimalist, you could try just adding a bit more sugar from time to time). You'll probably learn to feel what the pressure is like with a gentle squeeze, but this release mechanism (or 'spunding valve') means you shouldn't get into too much trouble.
I have tested it under water to see if there are any leaks, and after a bit of trial and error getting the backpressure right it seems to work perfectly. You need to make sure that your lid and tap is tight, and you might need to file down any moulding burrs to help them seal properly. You might also need to add an extra rubber washer to your tap, so you can keep it tight and vertical at the same time.
I tried a couple of different types of taps, and both the common white tap (with the 'B' shaped handle) and the black one (with the teardop-shaped handle) seem to handle the pressure just fine. I have also tried a yellow spring-top tap but it has a tendency to drip very slowly - not sure whether this would be universal or just mine (edit: randyrob seems to have the same type of tap in red and it looks lke it's OK).
Haven't served a brew through it yet, as I haven't got a bike inflater (I was using compressed air) but hopefully sometime in the next couple of weeks I'll be able to give it a trial run with real beer.
So we could be talking about a kegging (casking? cubing?) system for the cost of a cube, a gas dispenser, some parts and maybe an hour of farting around. By my reckoning it should be good for up to about 2 volumes of carbonation, but in reality it would probably be nearer 1.5 given that you wouldn't want to keep it permanently at its maximum pressure. I think that's still perfectly drinkable for day-to-day drinking (unless you like really fizzy lagers), and for the price I reckon it has to be worth a shot.
Beats bottling :beer: