We study statistical properties of packings of monodisperse spheres in a flat box. After "gravitational" filling and appropriate agitation, a nearly regular (in plane) but frustrated (normal to the plane) triangular lattice forms, where beads at individual sites touch either the front or back wall. It has striking analogies to order in antiferromagnetic Ising spin models. When tilting the container, Earth's gravitational field mimics external forces similar to magnetic fields in the spin systems. While packings in vertical containers adopt a frustrated state with statistical correlations between neighboring sites, the configurations continuously approach the predictions of a random Ising model when the cell tilt is increased. Our experiments offer insights into both the influence of geometrical constraints on random granular packing and a descriptive example of frustrated ordering.