Hot carriers in plasmonic nanostructures, generated via plasmon decay, play key roles in applications such as photocatalysis and in photodetectors that circumvent bandgap limitations. However, direct experimental quantification of steady-state energy distributions of hot carriers in nanostructures has so far been lacking. We present transport measurements from single-molecule junctions, created by trapping suitably chosen single molecules between an ultrathin gold film supporting surface plasmon polaritons and a scanning probe tip, that can provide quantification of plasmonic hot-carrier distributions. Our results show that Landau damping is the dominant physical mechanism of hot-carrier generation in nanoscale systems with strong confinement. The technique developed in this work will enable quantification of plasmonic hot-carrier distributions in nanophotonic and plasmonic devices.