Experimental studies show that porous sedimentary rocks have a fractal pore-grain interface. The bidisperse ballistic deposition model (BBDM) generates a porous structure by computer simulation. The model is able to reproduce the behavior of several macroscopic properties of sedimentary rocks quite well. In the present paper we show that the microstructure of the pore space created in BBDM is also realistic. Two-point correlations of the pore space agree qualitatively with results for real rocks, and the sample spanning pore has a fractal dimension in the range 2.29--2.99 for different porosities. We also study diffusion in the pore space by a random walk and show that it is anomalous at low porosities.