We put constraints on the properties of the progenitors of peculiar calcium-rich transients using the distribution of locations within their host galaxies. We confirm that this class of transients do not follow the galaxy stellar mass profile and are more likely to be found in remote locations of their apparent hosts. We test the hypothesis that these transients are from low-metallicity progenitors by comparing their spatial distributions with the predictions of self-consistent cosmological simulations that include star formation and chemical enrichment. We find that while metal-poor stars and our transient sample show a consistent preference for large offsets, metallicity alone cannot explain the extreme cases. Invoking a lower age limit on the progenitor helps to improve the match, indicating these events may result from a very old metal-poor population. We also investigate the radial distribution of globular cluster systems, and show that they too are consistent with the class of calcium-rich transients. Because photometric upper limits exist for globular clusters for some members of the class, a production mechanism related to the dense environment of globular clusters is not favoured for the calcium-rich events. However, the methods developed in this paper may be used in the future to constrain the effects of low metallicity on radially distant core-collapse events or help establish a correlation with globular clusters for other classes of peculiar explosions.