Coherent quantum information experiments are invariably performed in the non-relativistic limit with qubits whose dynamics are described by the Schrodinger equation. Accordingly, the Dirac equation of relativistic quantum mechanics cannot be realized in the laboratory without a method of mapping Dirac dynamics to Schrodinger dynamics. This has been achieved in recent years for the Dirac equation in 1+1 dimensions using trapped ions and Bose-Einstein condensates, while electrons in graphene obey the Dirac equation in 2+1 dimensions. Here, we show for the first time how the full 3+1 dimensional Dirac equation can be emulated with a non-relativistic four-level system with the level structure of a pair of coupled qubits. The system is manipulated by four classical resonant drives which determine the mass and momentum of the Dirac particle and any electric or magnetic fields, and the dynamics of the system are interpreted in terms of relativistic effects such as Zitterbewegung and Schwinger pair production. Some of these effects may accordingly be translated into the language of quantum optics. Our proposal is straightforward to realize with a superconducting Josephson circuit architecture.