Context. The Class 0 protostellar binary IRAS 16293–2422 is an interesting target for (sub)millimeter observations due to, both, the rich chemistry toward the two main components of the binary and its complex morphology. Its proximity to Earth allows the study of its physical and chemical structure on solar system scales using high angular resolution observations. Such data reveal a complex morphology that cannot be accounted for in traditional, spherical 1D models of the envelope. Aims. The purpose of this paper is to study the environment of the two components of the binary through 3D radiative transfer modeling and to compare with data from the Atacama Large Millimeter/submillimeter Array. Such comparisons can be used to constrain the protoplanetary disk structures, the luminosities of the two components of the binary and the chemistry of simple species. Methods. We present ¹³CO, C¹⁷O and C¹⁸O J = 3–2 observations from the ALMA Protostellar Interferometric Line Survey (PILS), together with a qualitative study of the dust and gas density distribution of IRAS 16293–2422. A 3D dust and gas model including disks and a dust filament between the two protostars is constructed which qualitatively reproduces the dust continuum and gas line emission. Results. Radiative transfer modeling in our sampled parameter space suggests that, while the disk around source A could not be constrained, the disk around source B has to be vertically extended. This puffed-up structure can be obtained with both a protoplanetary disk model with an unexpectedly high scale-height and with the density solution from an infalling, rotating collapse. Combined constraints on our 3D model, from observed dust continuum and CO isotopologue emission between the sources, corroborate that source A should be at least six times more luminous than source B. We also demonstrate that the volume of high-temperature regions where complex organic molecules arise is sensitive to whether or not the total luminosity is in a single radiation source or distributed into two sources, affecting the interpretation of earlier chemical modeling efforts of the IRAS 16293–2422 hot corino which used a single-source approximation. Conclusions. Radiative transfer modeling of source A and B, with the density solution of an infalling, rotating collapse or a protoplanetary disk model, can match the constraints for the disk-like emission around source A and B from the observed dust continuum and CO isotopologue gas emission. If a protoplanetary disk model is used around source B, it has to have an unusually high scale-height in order to reach the dust continuum peak emission value, while fulfilling the other observational constraints. Our 3D model requires source A to be much more luminous than source B; L_A ~ 18 L_⊙ and L_B ~ 3 L_⊙.