We report $^{75}$As- and $^{51}$V-nuclear magnetic resonance (NMR) measurements on the iron-based superconductor Sr$_2$VFeAsO$_3$ with alternating stacks structure. We find that the $^{75}$As nuclear spin-spin relaxation rate ($1/T_2$) shows a pronounced peak at $T_N$ = 165 K, below which the resonance peak shifts to a higher frequency due to the onset of an internal magnetic field. The $^{51}$V spectrum does not shift, but is broadened below $T_N$. We conclude that the Fe electrons oder antiferromagnetically below $T_N$ with a magnetic moment $m_{Fe}$ $sim$ 0.4 $mu_B$. Application of external pressure up to 2.4 GPa reduces $T_N$ in a rate of $-$40 K/GPa, and enhances the superconducting transition temperature $T_c$ in a rate of 2 K/GPa. The pressure-temperature phase diagram for Sr$_2$VFeAsO$_3$ shows that superconductivity coexists with antiferromagnetism over a wide pressure range with an unprecedented high $T_c$ up to 36.5 K.
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