I am curious, would it be possible to fabricate a modern BJT/FET transistor with hobby/garage tools only if size and aesthetics was not important?
Which materials would one use and how would one acquire them?
Well, yes and no.
You can make a transistor by hand in your kitchen. It will involve some nasty chemicals, but it is doable. A friend of mine made a LED a couple of years ago. Not a transistor, sure, but the same process was involved. (She was studying semiconductors at that time, so she had help. The process itself was quite similar to producing transistors).
However, It is very unlikely that you'll ever be able to make a transistor in your kitchen that even comes close to modern mass produced transistors. You wouldn't even come close to what has been produced in the 70th.
Nonetheless you may end up with a working transistor that does amplification. If you want to do this, and you're willing to invest some time and money into it, please do so! That would be really cool.
Sure it can be done, if you have ninja skillz:
I wanted to post more links but I don't have enough reputation points to allow that. Just search for the followup videos to the 2 links I have provided.
A very similar question has been asked before.
My answer there included this link to the website of a someone who has actually made thin film transistors out of zinc oxide.
That site also includes links to otehr people doing similar work - at home and in the garage.
So, it is possible to make transistors at home, but from what I see it is more of a learning experience as the produced transistors have really poor performance.
It's possible, but not so easy and the result is not going to be anywhere near the quality of components that you can buy. The main issue is that in order to actually form the semiconductor junctions inside of the transistor, you need to carefully change the properties of a piece of semiconductor in a controlled manner. This requires rather specialized tools as well as some pretty nasty chemicals. Namely, you at least need a quite high temperature (1000C) oven. See https://www.youtube.com/watch?v=w_znRopGtbE for a high level explanation of what you would need to do to make a FET.
To make something even remotely 'modern', you basically have to build your own clean room and acquire used semiconductor manufacturing equipment.
It is possible to make some very primitive diodes, such as what is used in a crystal radio, but this is more 'finding' a diode than making one as the actual junction is already in the crystal, you just need to go poking around with a piece of wire until you find one.
Your question is wrong, because size is one of the key attributes that defines a "modern" BJT or MOSFET.
Given that you really mean "is it possible to make a BJT or MOSFET using obsolete techniques in your garage" I would say it would be so difficult as to be almost impossible both from a technical and financial standpoint.
Using just used, refurbished equipment you would at the very least need a diffusion furnace, a spinner to apply resist, a few masks, a light source for photolithography, and some way to metallize the wafer (presumably using an evaporator). You'll also need a bath for wet etch. Then you have to have the ability to test your structures (they won't work for a while...) Good luck getting all that to work.
In addition, this all needs to be done in a clean environment, so you'll have to seal your garage and set up a laminar air flow and install air scrubbers.
This could EASILY run over $150k even using surplus gear. It just isn't a good hobby. There are a lot of fun things you can do in electronics. Transistor fabrication isn't one of them.
You could make capacitors or resistors or inductors. That could be fun.