Although the rocket-based X-ray Quantum Calorimetry (XQC) experiment was designed for Xray spectroscopy, the minimal shielding of its calorimeters, its low atmospheric overburden, and its low-threshold detectors make it among the most sensitive instruments for detecting or constraining strong interactions between dark matter particles and baryons. We use Monte Carlo simulations to obtain the precise limits the XQC experiment places on spin-independent interactions between dark matter and baryons, improving upon earlier analytical estimates. We find that the XQC experiment rules out a wide range of nucleon-scattering cross sections centered around one barn for dark matter particles with masses between 0.01 and 105 GeV. Our analysis also provides new constraints on cases where only a fraction of the dark matter strongly interacts with baryons.PACS numbers: 95.35.+d, 29.40.Vj