Following up Jeri Ellsworth’s home chip fab:

Building up the metal layers.
She currently uses conductive ink (the same one used to repair PCBs and hand-tune RF microstrip lines). This can be done with a stencil but the reliability and resolution are low. Furthermore, this ink is very expensive.
What is typically done in the industry to build such layers with a high reliability and resolution is a physical vapor deposition process. The physics behind it is simple: you just boil off a metal (typ. aluminium) by heating it (aluminium as a pretty low melting point), and condense it on the (colder) substrate. You make a vacuum to remove impurities, increase the mean free path of the aluminium atoms, and further lower the melting point.
For a good quality deposition, the required level of vacuum is high. The industry uses turbomolecular pumps, which are fast, but also extremely expensive, delicate, fragile (if for some reason the vacuum chamber is brutally open, the pump is irreversibly destroyed) and hard to deal with. However, for slower pumping speed (and therefore more time to make the vacuum), oil diffusion pumps are suitable. They are cheaper and harder to destroy. Some are available from eBay for a few hundred euros.
Both models of pumps do not operate at atmospheric pressure and require a traditional (rotary vane) vacuum pump in series. Several people told me that refrigerator compressors will not work (pressure drop is not low enough + the pump will fail when operated in air for a long time). Air conditioner pumps might do better.
For melting the aluminium, I have seen a design that pumps high currents (around 300A) through a tungsten crucible. While it is very possible to replicate this design (FYI we have a 3.3V 1000A power supply at the lab, otherwise I guess welding stations are fine) any other heating system that can heat a few dozens grams of aluminium to 600-700 degrees in vacuum is OK.
Having this physical vapor deposition system will not only enable metallization of chips in the long term, but also build-up of PCB-like devices on various substrates (glass, soft plastics, …) in the shorter term.

Exposing the photoresist with high precision.
The industry typically uses a special kind of expensive mercury vapor lamp to get the appropriate wavelength at a relatively high power (UV tubes for PCBs, UVPROMs and night clubs are apparently NOT suitable). But nowadays this is sometimes replaced with much cheaper and reliable UV LEDs.
To increase resolution, it is tempting to print an enlarged mask of the chip and then use an optical system to scale it down to chip size. However, the problem here is aligning the different layers. Maybe a high quality optical photo scaling machine mounted backwards with a microscope plus micrometers could work. The idea is to align the chip under visible light and control with the microscope, then switch on the UV light for photoresist exposure. The problem we might have is that the lenses can behave significantly differently at the two wavelengths, but maybe this could be compensated for.