Although signatures of superconductivity in Dirac semimetals have been reported, for instance by applying pressure or using point contacts, our understanding of the topological aspects of Dirac semimetal superconductivity is still developing. Here, we utilize nanoscale phase-sensitive junction technology to induce superconductivity in the Dirac semimetal Bi1-xSbx. Our radiofrequency irradiation experiments then reveal a significant contribution of 4π-periodic Andreev bound states to the supercurrent in Nb-Bi0.97Sb0.03-Nb Josephson junctions. The conditions for a substantial 4π contribution to the supercurrent are favourable because of the Dirac cone's very broad transmission resonances and a measurement frequency faster than the quasiparticle poisoning rate. In addition, we show that a magnetic field applied in the plane of the junction allows tuning of the Josephson junctions from 0 to π regimes. Our results open the technologically appealing avenue of employing the topological bulk properties of Dirac semimetals for topological superconductivity research and topological quantum computer development.