Do they still have PC kits available? I know heathkit had one at one point, years ago. Something very low-end, like an 8080 with 4k memory. If not a kit, how about instructions to build one with currently available parts?
By "build one", do you mean (1) mounting a motherboard and drives into a cabinet and pluging in some memory chips and power supply OR (2) soldering individual parts and sockets onto a large piece of circuitboard?
I remember that Sinclair (or similar name) made a kit. I presume that it would run BASIC but that might be wrong.
My observation: a low end PC contains an awful lot of parts leaving a gazillion or more opportunities for error if wired by hand - would be fairly large too. I've not seen any kits in years - in terms of solder parts on board kind.
I am not sure what you intend to do but microcontrollers seem to be within reach of many hobbyists - PICs and BASIC STAMPs might be some examples of these - though there are others. You might want to take a look at this stuff - it may be what you are looking for. It appears that PICs and similar microcontrollers run machine type language that is compiled on a PC then downloaded for execution by the micro and companion memory. A BASIC STAMP, as I understand it, contains the lines of BASIC code on board. The nice thing is that the parts are low cost. I have no experience (yet) but the technology is robust enough for even the most basic experimenter or hobbyist.
I built a Sinclair ZX81 kit [back in 1980], but I doubt they still exist.
You could get a microcontroller experimenter board from places like: http://www.melabs.com/ http://www.parallax.com/ http://www.basicmicro.com/
If you want a chip similar to the 8080,
Jameco still sells the 8052BASIC micro controller. http://www.jameco.com/
An excellent book about the 8052 is Jan Axelson's
The Microcontroller Idea Book **broken link removed**
Not for newbies but my fav is http://www.newmicros.com/
they have a 8051 kit that is only $40.