Making a homebrew transistor should be possible, it probably would be quite expensive and time-consuming and the final product would be quite crappy, but it would be cool.
I would start with a diode though, since it is a simpler device. Getting a n+ doped silicon wafer from a nearby university lab should be possible, if it isn't considered cheating. An EE friend of mine has a whole 10 cm wafer of Schottky diodes from a lab he did once, so it can't be too expensive. I really don't know how to make a Schottky diode, but I think it is just a silicon-metal junction, so I'll consider a normal pn-junction diode, although a Schottky diode maybe much simpler.
Having a n+ substrate, one would need to p dope the surface with Boron in a diffusion furnace. In the garage maybe a pyrex tube in a kiln would do the trick. First of all the parts of the wafer which isn't supposed to be doped have to be masked, maybe a cut piece of glass would do. The wafer is placed inside the tube, which is sealed and has a supply of inert gas and Boron connected to it. The kiln should be set to around 1000 °C and it should probably be left on for more than 4 hours. I don't think overdoing it would be too bad.
Then , there's the contacts. Vaporizing aluminum in a vacuum chamber and allowing it to condense over the (masked) wafer should make adequate contacts. A good vacuum would be needed, roughing with a mechanical pump and finishing with a sorption pump should do the trick.
Finished. Does it work? Probably not. Still, something that I just have to try out sometime.
Oops, this turned into a general semiconductor processing writeup. Anyway, this is a ASCII drawing of what a finished diode would look like:
! \ p doped / !
! \___________/ !
! n+ substrate !
Sorry, this was my first attempt at ASCII drawings in HTML.