We investigate interfacial and capillary phenomena in a simple model for a fluid of hard rods, viz. the Zwanzig model, in which the orientations of rectangular blocks are restricted to three orthogonal directions. The theory, which is based on an Onsager-like free energy functional, predicts local biaxial ordering at the "free" interface between the coexisting isotropic and nematic phases. For an isotropic bulk fluid in contact with a single planar hard wall, we find a continuous surface phase transition from uniaxial to biaxial local symmetry, followed by complete wetting of the wall-isotropic fluid interface by a nematic film with director parallel to the wall, as the reservoir density approaches its value at bulk coexistence. For a fluid confined by two parallel hard walls we determine a first-order capillary nematization transition at large wall separation, which terminates in a capillary critical point when the wall separation is about twice the length of the rods. This transition is the analog of the capillary condensation observed for simple fluids confined by attractive walls but is purely entropy driven here.