We perform molecular dynamics simulations to compress binary hard spheres into jammed packings as a function of the compression rate R, size ratio alpha, and number fraction x(S) of small particles to determine the connection between the glass-forming ability (GFA) and packing efficiency in bulk metallic glasses (BMGs). We define the GFA by measuring the critical compression rate R-c, below which jammed hard-sphere packings begin to form "random crystal" structures with defects. We find that for systems with alpha >= 0.8 that do not demix, R-c decreases strongly with Delta phi(J), as R-c similar to exp(-1/Delta phi(2)(J)), where Delta phi(J) is the difference between the average packing fraction of the amorphous packings and random crystal structures at R-c. Systems with alpha <= 0.8 partially demix, which promotes crystallization, but we still find a strong correlation between R-c and Delta phi(J). We show that known metal-metal BMGs occur in the regions of the alpha and x(S) parameter space with the lowest values of R-c for binary hard spheres. Our results emphasize that maximizing GFA in binary systems involves two competing effects: minimizing alpha to increase packing efficiency, while maximizing alpha to prevent demixing.