One low-tech way to measure the thickness of very thin wire if you don’t have a micrometer is to use a ruler on a flat surface (sheet of glass maybe?), and something of a known thickness several times as thick as the wire. (a drill or coin perhaps?).
Put the drill or coin on the flat surface and hold the ruler with one end on the it at the 300mm point and the other on the glass, so there is a very narrow triangular gap between the ruler and the surface. Slide the wire into the wide end of the gap and move it along towards the thin end until it sticks. Read off the distance on the ruler. Say it is 56mm and the coin is a newish 2p piece, which is 2.08mm thick.
Checking the wire with a micrometer shows an actual diameter of 0.375mm, about 4% error.