He chained rough guesses: population, pianos per household, and tunings each tuner can do per year
Fermi assumed about 3 million people, roughly one piano per 100 households, each piano tuned once a year, and one tuner handling a few pianos a day for most of the year. Multiplying it out gives a few dozen to a hundred tuners — close enough to the real count to be uncannily useful.
