The discovery of a two-dimensional electron gas (2DEG) at the interface between insulating oxides has led to a well-deserved level of excitement due to possible applications as ``in-plane'' all-oxide nanoelectronics. Here we expand the range of possibilities to the realm of ``out-of-plane'' nanoelectronics by examining such all-oxide heterostructures as barriers in tunnel junctions. As an example system we perform first-principles electronic structure and transport calculations of a tunnel junction with a ${[{\text{SrTiO}}_{3}]}_{4}/{[\text{LaO}]}_{1}/{[{\text{SrTiO}}_{3}]}_{4}$ heterostructure tunneling barrier embedded between ${\text{SrRuO}}_{3}$ electrodes. The presence of the LaO atomic layer induces the formation of a 2DEG within the tunneling barrier which acts as an extended 2D potential well perpendicular to the transport direction, providing a route for resonant tunneling. Our calculations demonstrate that the tunneling conductance in this system can be strongly enhanced compared to a pure ${\text{SrTiO}}_{3}$ barrier due to resonant tunneling, but that lattice polarization effects play a significant role in determining this behavior. In addition we find that this resonant tunneling is highly selective of the orbital symmetry of the tunneling states due to the ``orbital polarization'' of the 2DEG. We also discuss how the properties of the 2DEG are affected by the presence of metal electrodes.