Are the terms “gnat's eye” and “hair's breadth” true units of measure?

Yes, but both are slang units of distance and diameter. The eyes of typical gnats tend to have diameters similar in size to a hair’s breadth—roughly 100–150 micrometers. An item would have to be very short in order to be gnat’s eye in length! A hair’s breadth is an informal unit of distance: it is used to denote a measurement of approximately 70 to 100 micrometers in diameter, or 0.1 millimeter, which is similar in thickness to real human hair.