Download the project from github
Manufacture the PCB at your favourite PCB fab (I recommend elecrow.com). I've already included the gerbers so you don't even have to use eagle to export them - they are exported for use with elecrow so you may need to export them again if you want to manufacture them somewhere else.
Order the necessary parts listed in bill of materials (I recommend Farnell/Element 14). Nixies themselves can be bought at siberian-shop.com or somewhere else. Some of the things you probably already have or can be bought cheaper on ebay - like headers, power jacks, ...
Solder everything together. SMD soldering isn't hard if you are afraid of that. Using solder paste and hot air will probably be faster than a regular soldering iron.
Flash the firmware to the micro. You will need any STM development board (they can be bought at Farnell for less than 10€) with STLINK. You can compile the project using free EmBlocks IDE (now also called EmBitz). It's a GCC based IDE so it should work with other compilers/IDEs as well. If you don't want to compile the project then you can simply used the .hex file in the bin folder. It's configured for use without the temperature sensor. But compiling everything in EmBlocks should only take a few clicks so don't worry if you have never done it before.
To flash the firmware just connect 5 wires to the SWD connector on your development board, remove the two STLINK jumpers and flash the hex file using ST Link utility.
3D print the case. It's designed to be 3D printable on most popular home 3D printers. Don't like the case? Make one yourself.
Assemble everything, add a coin battery for the real time clock and enjoy