Theoretical work predicts that Josephson junctions containing metallic altermagnetic barriers should display 0-π transitions of the critical current as a function of both barrier thickness and temperature, with the decay and oscillation period of the supercurrent depending on the orientation of the crystal axes relative to the transport direction. Motivated by these predictions, and by reports of a compensated magnetic phase attributed to altermagnetism in epitaxial Mn₅Si₃ thin films, we fabricate and measure Nb/Pt/Mn₅Si₃/Pt/Nb Josephson junctions varying the thickness of the Mn₅Si₃ barrier. The critical current decays as a single exponential over more than four orders of magnitude with decay length ξ_(Mn₅Si₃) = 0.31 ± 0.03 nm, shorter than reported for Josephson junctions containing the metallic antiferromagnets FeMn, Cr, and NiMn. The Mn₅Si₃ barrier has an estimated current-perpendicular-to-plane resistivity of 320 ± 10 μΩ cm. No 0-π transition is resolved at the sampled barrier thicknesses, and the temperature dependence of the critical current of a junction with a 1 nm barrier is smooth and monotonic. We discuss the absence of resolvable transitions in terms of the microstructure of the barrier, its uncertain magnetic phase, and the narrow thickness window imposed by the rapid decay, and identify barriers with well-defined crystalline orientation as the key requirement for future tests of altermagnetic Josephson physics.