Our calculation shows that negative refractive index (NRI), which was known to exist only in metamaterials in the past, can be found in Dirac semimetals (DSM). Electrons in DSM have zero effective mass and hence the system carries no nominal energy scale. Therefore, unlike those of ordinary materials, the electromagnetic responses of the electrons in DSM will not be overwhelmed by the physical effects related to electron mass. NRI is induced by the combination of the quantum effect of vacuum polarization and its finite temperature correction, which is proportional to T^{4} at low temperature. It is a phenomenon of resonance between the incident light and the unique structure of Dirac cones, which allows numerous states to participate in electron-hole pair production excited by the incident light with a similar dispersion relation to that of Dirac cones. The NRI phenomenon of DSM manifests in an extensive range of photon frequencies and wave numbers and can be observed around the gigahertz range at room temperature.