Electrical contacts between nanoengineered systems are expected to constitute the basic building blocks of future nanoscale electronics. However, the accurate characterization and understanding of electrical contacts at the nanoscale is an experimentally challenging task. Here, we employ low-temperature scanning tunneling spectroscopy to investigate the conductance of individual nanocontacts formed between flat Pb islands and their supporting substrates. We observe a suppression of the differential tunnel conductance at small bias voltages due to dynamical Coulomb blockade effects. The differential conductance spectra allow us to determine the capacitances and resistances of the electrical contacts which depend systematically on the island-substrate contact area. Calculations based on the theory of environmentally assisted tunneling agree well with the measurements.