We present results of an experimental study on single crystals of a 5d1 double perovskite Ba2CaReO6. Magnetization measurements reveal a weak splitting between zero-field-cooled and field-cooled protocols below 12 K. At magnetic fields above 1 T the splitting is absent and the magnetic susceptibility is featureless. A detailed specific heat study in a wide temperature range and comprising different heat pulses did not reveal any indication of a thermodynamic phase transition. At low temperatures we do observe specific heat deviating from a phonon background, leading to a total electronic entropy release of ∼Rln2. Resonant and non-resonant x-ray diffraction of characteristic Bragg peaks indicates a significant presence of disorder, potentially related to random tilts and rotations of rigid ReO6 octahedra.