We investigate the I(V) characteristics (current versus bias voltage) of side-gated quantum-point contacts, defined in GaAs/AlxGa1-xAs heterostructures. These point contacts are operated in the closed-channel regime, that is, at fixed gate voltages below zero-bias pinch-off for conductance. Our analysis is based on a single scaling factor, extracted from the experimental I(V) characteristics. For both polarities, this scaling factor transforms the change of bias voltage into a change of electron energy. The latter is determined with respect to the top of the potential barrier of the contact. Such a built-in energy-voltage calibration allows us to distinguish between the different contributions to the electron transport across the pinched-off contact due to thermal activation or quantum tunneling. The first involves the height of the barrier, and the latter also its length. In the model that we are using the channel length remains the only adjustable parameter since the barrier height can be experimentally determined. For short (similar to 0.06 mu m) contacts, the I(V)-derived lengths agree rather well with those estimated from the geometrical layout, whereas nominally long (similar to 1.2 mu m) contacts are typically found to consist of very short (similar to 0.2 mu m) barriers. We have mapped the height of the barrier as a function of the gate voltage, and found that its behavior differs strongly from that extrapolated using conventional bias spectroscopy in the open-channel regime above conductance pinch-off.