Motivated by the recent discovery of near-room temperature superconductivity in high-pressure superhydrides, we investigate from first-principles the high-pressure superconducting phase diagram of the ternary Ca-B-H system, using ab-initio evolutionary crystal structure prediction, and Density Functional Perturbation Theory. We find that below 100 GPa all stable and weakly metastable phases are insulating. This pressure marks the appearance of several new chemically-forbidden phases on the hull of stability, and the first onset of metalization in CaBH$_5$. Metallization is then gradually achieved at higher pressure at different compositions. Among the metallic phases stable in the Megabar regime, we predict two high-$T_c$ superconducting phases with CaBH$_6$ and Ca$_2$B$_2$H$_{13}$ compositions, with critical temperatures of 119 and 89 K at 300 GPa, respectively, surviving to lower pressures. Ternary hydrides will most likely play a major role in superconductivity research in the coming years; our study suggests that, in order to reduce the pressure for the onset of metallicity and superconductivity, further explorations of ternary hydrides should focus on elements less electronegative than boron.
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