We study the electronic transport through epitaxial GaAs nanopillars that are only 16 nm long, with diameters of about 100 nm at the upper and 40 nm at the lower end. The pillars can be considered to be very short conical nanowires embedded in AlGaAs. They represent quantum point contacts between two perfectly lattice matched three-dimensional GaAs charge reservoirs. Distinctive asymmetries are found in the current-voltage characteristics. We associate them with the conical shape of the pillars. Although contact reservoirs and pillars are made from the same material, the transport through the pillars is dominated by tunneling across shallow barriers. This is explained by the quantum size effect on the electronic states within the pillars.