Context. Masses are one of the most difficult stellar properties to measure. In the case of the white-dwarf (WD) companions of Barium (Ba) stars, the situation is worse. These stars are dim, cool, and difficult to observe via direct methods. However, Ba stars were polluted by the Asymptotic Giant Branch (AGB) progenitors of these WDs with matter rich in heavy elements, and the properties of their WD companions contain key information about binary interaction processes involving AGB stars and about the slow neutron capture process (s-process) of nucleosynthesis.Aims. With this study, we aim to determine accurate and assumption-free masses for the WD companions of as many Ba stars as possible. We want to provide new observational constraints that can help us learn about the formation and evolution of these postinteraction binary systems and about the nucleosythesis processes that took place in the interiors of their AGB progenitors. Methods. We combined archival radial-velocity data with Hipparcos and Gaia astrometry using the software package orvara, a code designed to simultaneously fit a single Keplerian model to any combination of these types of data using a parallel-tempering Markov chain Monte Carlo method. We adopted Gaussian priors for the Ba star masses and for the parallaxes, and assumed uninformative priors for the orbital elements and the WD masses. Results. We determined new orbital inclinations and companion masses for 60 Ba star systems. These results include a couple of new orbits and several improved orbits for the longest-period systems. Additionally, we unravelled a new triple system that was not known before and constrained the orbits and the masses of the two companions. Conclusions. The WD mass distribution presented in this work is compatible with that of field WDs and with the distributions published before for Ba star companions. A few WD companions have masses higher than 0.8 M , considering 1-σ uncertainties. This indicates that they might come from AGB stars that are more massive than 3 M . These masses are higher than what the abundance ratios on Ba star atmospheres and theoretical models of the s-process of nucleosynthesis seem to expect, raising interesting questions about the formation of these systems.