Confined mixtures of a polymer and nonspecifically binding particles (condensers) are studied as models for viruses containing double-stranded DNA (polymer) and condensing proteins (particles). We explore a model in which all interactions between the packed content (polymer and particles) and its confinement are purely repulsive, with only a short-range attraction between the condensers and polymer to simulate binding. In the range of physical parameters applicable to viruses, the model predicts reduction of pressure in the system effected by the condensers, despite the reduction in free volume. Condensers are found to be interspersed throughout the spherical confinement and only partially wrapped in the polymer, which acts as an effective medium for the condenser interactions. Crowding of the viral interior influences the DNA and protein organization, producing a picture inconsistent with a chromatin-like, beads-on-a-string structure. The model predicts an organization of the confined interior compatible with experimental data on unperturbed adenoviruses and polyomaviruses, at the same time providing insight into the role of condensing proteins in the viral infectious cycles of related viral families.