What unit measures one billionth of a meter?

The unit that measures one billionth of a meter is the nanometer. This specific measurement is fundamental in the field of nanotechnology, where dimensions on the nanoscale, typically ranging from 1 to 100 nanometers, are critical for understanding materials and phenomena at the atomic and molecular level.

For context, the other terms listed do relate to the field of nanotechnology, yet they do not serve as units of measurement. The term "nanoscale" refers to the scale at which nanoscale phenomena occur but does not quantify sizes. "Nanoparticles" refers to small particles within the nanometer range but does not specify a measurement itself. "Nanorobots" are theoretical machines designed at the nanoscale but again do not represent a unit of measurement. Therefore, "nanometer" is the correct answer as it specifically denotes a measurement of length equal to one billionth of a meter.