Go watch some videos of Kern machines. Their demonstrations of drilling through human hairs with a VMC are pretty compelling.

it might be better to say that machine components and coordinate measuring systems have become cheaper, so that high accuracy is more attainable.

Good machines 50 years ago could hold .001" tolerances with care. Most machines nowadays can do that without that much care. Some machines nowadays can hold .0001" tolerances with care, and some can do even better.

If you don't limit the way of machining, you can count microchip manufacturing as machining. So, we can easily talk about nanometers.

If you consider machining only physically removing the material by friction of two different materials, then I think it is practically about 0.001mm. Even this dimension is not always possible. because in this level of precision requires to consider even ambient temperature. If you machine the material with a cold coolant and wait for 10 mins to measure the result. You would end up a bigger part. If you machine it under a warm coolant, then you would end up a smaller part.

it depends on how much you're willing to spend, and in some areas, whether your company is allow to import specific devices which are highly precise.

Many of the most challenging precision and accuracy problems were well understood and documented quite some time ago (books like Moore's Foundations of Mechanical Accuracy). Even today with DIY devices, people are getting impressive results, but often it's hard to say for sure just how accurate something is without expensive calibration tools and custom built environments. For example, if you're working with high precision machining, the temperature of the room matters (since the work will expand or contract) so you will probably need a climate controlled room inside a climate controlled room, etc. Machine needs to be rigid, feedback control, sophisticated optical sensing for position, all makes this very expensive.

I think the folks who make large glass mirrors do well under 0.5 micron accuracy.

As precise as you are willing to pay, usually micron resolution is the limit:

https://www.youtube.com/watch?v=2dsrLD52Mv0

This is a DIY EDM(Electrical discharge machine). Proffesional machines are much better (and more expensive too).

What is good about EDM is that it doesn't care about the hardness of the material you work with , only about its conductivity. You can shape diamond or the strongest steel without problems.

terms like thousandths of an inch are thrown around like it's nothing in machine shops, so I'd say pretty damn precise.

That kind of precision was commonplace 40 years ago, not just today.

They have, but they're also still quite expensive.