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Fist at arms length = ~ 10 arc degrees Thumb at arms length = ~ 2 arc degrees Little finger ~ 1 arc degree
ONE INCH but depends on the people
You do not need "white rods", you need a tape measure or a ruler to measure the lenght of your thumb.
About 1.14 inches, say, about 1/2 the length of your thumb
It depends on ewhat attribute of the thumb you wish to measure: Some attributes of possible interest are length, mass, conductivity, tensile strength. If length, probably centimetre. Millimetre or smaller if you wanted an absurd degree of precision.