We performed fixed-node diffusion Monte Carlo (DMC) calculations to investigate structural and energetic properties of graphenylene (GPNL), a two-dimensional network of sp2-bonded carbon atoms with large near-circular pores, and its H2 separation performance for gas mixtures. We have found that the energetic stability of a GPNL monolayer is comparable to that of γ-graphyne, as evidenced by its large cohesive energy of 6.755(3) eV/atom. Diffusion barriers of several gas molecules, including hydrogen, through a GPNL membrane were determined from the analysis of their adsorption energies depending on the adsorption distance, which led to our estimation for hydrogen selectivity with respect to other target molecules. DMC hydrogen selectivity of a GPNL monolayer was found to be exceptionally high at 300 K, as high as 1010-1011 against CO and N2 gases. This, along with high hydrogen permeance due to its generic pore structure, leads us to conclude that GPNL is a promising membrane to be used as a high-performance hydrogen separator from gas mixtures. We find that when compared to our DMC results, DFT calculations tend to overestimate H2 selectivity, which is mostly due to their inaccurate description of short-range repulsive interactions.