[1]: https://conorpp.com/2016/03/13/my-experience-with-dirtypcbsc...
In open source designs you can usually spot OSH Park boards by the distinctive purple color. Seeed Studio and the other Asian budget board houses are a decent economy option if you can get away with looser tolerances. OSH Park is nice for compact designs because the gold-flashed pads are better for surface-mount parts, since they aren't raised like traditional solder-painted pads. Unlike most quickturn PCB fabs they don't bury you with options; the standard options give you everything you need (ENIG, double mask+silk screen, either 2/4 layer, etc.). The OSH Park ordering wizard is a case study in how PCB orders should be done (upload and a couple clicks, with a graphical preview), and for the quality the price can't be beat. Because orders are pooled and then separated before shipping out, it is slower.
You can use the free and open KiCad[2] software (again, what was used in OP; tutorials[3,4]) to do schematic capture and board layout, have it generate some gerber and drill files, and then order the boards for a few bucks from OSH Park. KiCad won't do simulations like Altium or some of the other commercial packages, but if you just want to take a schematic and make a board it works fine. Definitely learn its hotkeys.
My last company did care about turn time but also wanted affordable boards. They used PCB-Pool: http://www.pcb-pool.com/ppus/index.html
Not sure if these can be applied to digital circuits, perhaps not very easily, since a lot of components are SMD these days, at least in the high speed digital domain.
The most economical stencils are avaliable from OSH Stencils. They offer mylar stencils for $0.64/square inch, with a minimum charge of $5. Mylar stencils are perfectly satisfactory for prototyping or even short production runs.
On that note, the title neglects the price of the board, which will add to the total cost for building this. From OSH Park it's $4.60 for three copies of the board, bringing the total per token to $4.50 + $1.53 = ~$6.
That's a very interesting microcontroller by the way, I always think it's mildly amusing with 8-bit CPU cores sitting on a hardware USB peripheral. Looks like a nice chip, plenty of I/O, 5 V-capable, built-in 3.3 V regulator and USB firmware, and stuff. Thanks OP for sharing this!
That being said, I suspect that methodology doesn't scale down to such small features, multiple layers etc. that you see in the link you posted, so at some point you have to give up on DIY etching.
It's quite a shame the "fablabs" have chosen to go with that approach (for philosophical reasons mostly, I guess, as the milling machine could also be used for other purposes) rather than a decent simple etching process.
Sorry for the rant, I've seen a "fab academy" student struggle with the process quite unreasonably during the last few weeks...
http://www.instructables.com/id/T962A-SMD-Reflow-Oven-FixHac...
I have the T962A and it works really well with the mods. I generally reflow batches of 12 boards at a time (on custom manufacturing jigs). Much easier than hand-soldering and the results are near-perfect.