Multicellular cable bacteria display an exceptional form of biological conduction, channeling electric currents across centimeter distances through a regular network of protein fibers embedded in the cell envelope. The fiber conductivity is among the highest recorded for biomaterials, but the underlying mechanism of electron transport remains elusive. Here, we performed detailed characterization of the conductance from room temperature down to liquid helium temperature to attain insight into the mechanism of long-range conduction. A consistent behavior is seen within and across individual filaments. The conductance near room temperature reveals thermally activated behavior, yet with a low activation energy. At cryogenic temperatures, the conductance at moderate electric fields becomes virtually independent of temperature, suggesting that quantum vibrations couple to the charge transport through nuclear tunneling. Our data support an incoherent multistep hopping model within parallel conduction channels with a low activation energy and high transfer efficiency between hopping sites. This model explains the capacity of cable bacteria to transport electrons across centimeter-scale distances, thus illustrating how electric currents can be guided through extremely long supramolecular protein structures.