A DIY, 8 layers PCB, desktop machine?
That's not possible with a single machine, that's a factory.
I previously missed the goals of 8 layers and 0201 part size. I would not say it's not possible, but those are tough goals to meet. I do believe it' would be madness tho attempt this without a lot of experience from previous projects. Machines such as the voltera and other machines shown in that video are doable if you have some decent tools (lathe, milling machine) yourself. But building your first machine that does much more then those machines is very likely doomed to fail.
Some brainstorming ideas:
One thing I am curious about and think about every now and then is a prototype machine that uses enameled wire for the connections. It could start with dual layer FR4, keep one layer for GND, and the other for milling pads and some connections, but the majority of the connections is then made by a CNC machine that puts enameled wires though the holes, and cuts and bends the ends. As far back as the early '80-ies or maybe even the '70-ies there were automated wire wrapping machines and also some very unconventional other automated processes.
And I guess that answers my ramblings too. All those machines went away. When electronics became much smaller and frequencies increased, prices of outsourcing also decreased, and things like impedance matching became more important. Back then the only alternative was many hundreds of man hours (form real humans, which do not work without a salary (except for russian soldiers) in wire wrapping and these days you have to compete with reliable PCB prototyping services, that also create a PCB that is the same as a production PCB, so it can be used for EMC testing etc.
Another quite different process that may (or not) be adaptable to "desktop size" is LtCC (Low Temperature Cofired Ceramics). I never looked into the exact details of how that process works, but I do know that is has a very good resolution and accuracy, builds up the product one layer at a time and can have many layers (over 50).
There are also other less mainstream processes that may be adoptable. For example there are metal core PCB's that have multiple layers of conductors laminated on top. I also do not know how those are made. Maybe you can get inspiration on how flex PCB's are made.
Using some double sided sticky tape would be an intriguing idea. First you start with sticky layer, and then you push in bare (not enameled) copper wire into the glue, and you cover it with the next layer of tape. Maybe it's doable, but it would be tough to get this to work reliably, especially on fine pitch stuff, and there is still the detail of how to do pads.
But whatever direction you take, at such resolutions it will become a quite expensive machine and being made for a small niche market, and that is not a good combination.