We present the results of a survey for stellar and substellar companions to 82 young stars in the nearby OB association Upper Scorpius. This survey used nonredundant aperture mask interferometry to achieve typical contrast limits of ΔK ∼ 5-6 at the diffraction limit, revealing 12 new binary companions that lay below the detection limits of traditional high-resolution imaging; we also summarize a complementary snapshot imaging survey that discovered seven directly resolved companions. The overall frequency of binary companions (∼35-4 +5% at separations of 6-435 AU) appears to be equivalent to field stars of similar mass, but companions could be more common among lower mass stars than for the field. The companion mass function has statistically significant differences compared to several suggested mass functions for the field, and we suggest an alternate lognormal parameterization of the mass function. Our survey limits encompass the entire brown dwarf mass range, but we only detected a single companion that might be a brown dwarf; this deficit resembles the so-called brown dwarf desert that has been observed by radial velocity planet searches. Finally, our survey's deep detection limits extend into the top of the planetary mass function, reaching 8-12 M Jup for half of our sample. We have not identified any planetary companions at high confidence (≲99.5%), but we have identified four candidate companions at lower confidence (≲97.5%) that merit additional follow-up to confirm or disprove their existence.