Would it be possible to make a solid state transistor by hand?
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.