How do you make a machine that has perfectly flat/plumb/level metal components without the aide of a machine to begin with?
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Geometry, optics, and abrasion.
Rub two surfaces together. The high spots of one will get rubbed off by the high spots of the other until they come into perfect intimate contact everywhere.
There are only three geometric surfaces that remain in contact everywhere as you slide them across each other: planes, spheres, and cylinders. So now you can make perfect flat sheets, spheres, straight edges and circles just by rubbing things together.
Once you have a straight edge, other tricks from geometry allow you to divide a line perfectly in half, draw perfect triangles and squares, etc. The law of similar triangles allows you to build things like the [pantograph](https://en.m.wikipedia.org/wiki/Pantograph) and other tools that can duplicate a shape at half the size and twice the accuracy.
Light travels in an (almost) perfectly straight line. So just by lining up objects visually (e.g. a gunsight), you can create perfectly straight lines of arbitrary length, and generally do all the geometric tricks above to create and copy shapes on very large or very small scales. If you need better precision than the human eye, you can use the rubbing trick to create spherical lenses, and thus magnifying glasses, microscopes and telescopes.
Light is a wave, with crests and troughs less than a millionth of a meter apart. Two light waves can be made to interfere, canceling each other out if the crests of one line up exactly with the troughs of the other. If those waves are reflected from two different surfaces, this lets you measure the distance between them to within a fraction of a wavelength of light — a precision of tenths of millionths of a meter!
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