Using a lathe as an example, I have access to a lathe that is precise down to 0.0002 inches. This lathe contains various cylindrical parts that had to have been machined by something at least that precise. While I have no doubt we have lathes which are more precise today, something had to make those ones, and the ones before that, and so on until you run out of more precise lathes. Similarly, how did we make measuring tools precise enough to tell us how imprecise our manufacturing tools are?
Tl;dr: Tools started out as rocks and copper accurate down a handful of millimeters, and now we can measure nanometers. How did we bridge that gap?
This is more of a question for Archaeologists or Anthropologists of culture, but let me give you some examples.
Since at least Early Medieval age, foot powered lathes were commonplace. A pole lathe is a very crude tool that requires only simplest iron elements any blacksmith could make, and some crudely chopped wood, yet can be used to create near perfect toruses, globes, wheels, circular joints etc. It is not an exaggeration that a lathe is called "the mother of all machines". Being able to make a (near) perfectly circular object and a basic knowledge of practical geometry/maths (not school-learned, just trial and error and apprenticeship) opens up the possibility to make just about any tool
With just as much simplicity, a medieval or Ancient craftsman could make a crude caliper, compass, or a measuring rod with relatively precise step.
With those above and some careful thinking, one would be able to design almost every human-powered tool possible with the previous set of tools, and use these second generation tools to make even more precise ones etc etc
Precision is just measurement and repetition. It does not even matter how precise is your initial tool and measurement, as long as it can be repeated many times and then physically averaged out. Measuring rulers, calipers, compasses, trysquares, and every imaginable hand measuring tool under the sun were already in use in Antiquity, they just used different unit of measurement. Of course, before the 18th century those tools could not be much more precise than a human eyesight would allow (even with the limited use of magnifying lenses).
Suggested reading:
Klein, Herbert A. The science of measurement: a historical survey
Whitelaw, Ian (2007). A Measure of All Things: The Story of Man and Measurement.