We report a $mu$SR study on the antiperovskite oxide superconductor Sr$_{3-x}$SnO. With transverse-field $mu$SR, we observed the increase of the muon relaxation rate upon cooling below the superconducting transition temperature $T_{mathrm{c}}=5.4$ K, evidencing bulk superconductivity. The exponential temperature dependence of the relaxation rate $sigma$ at low temperatures suggests a fully gapped superconducting state. We evaluated the zero-temperature penetration depth $lambda(0)propto1/sqrt{sigma(0)}$ to be around 320-1020 nm. Such a large value is consistent with the picture of a doped Dirac semimetal. Moreover, we revealed that the ratio $T_{mathrm{c}}/lambda(0)^{-2}$ is larger than those of ordinary superconductors and is comparable to those of unconventional superconductors. The relatively high $T_{mathrm{c}}$ for small carrier density may hint at an unconventional pairing mechanism beyond the ordinary phonon-mediated pairing. In addition, zero-field $mu$SR did not provide evidence of broken time-reversal symmetry in the superconducting state. These features are consistent with the theoretically proposed topological superconducting state in Sr$_{3-x}$SnO, as well as with $s$-wave superconductivity.
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