In this work, we examine how volume exclusion caused by regions of high chromatin density might influence the time required for proteins to find specific DNA binding sites. The spatial variation of chromatin density within mouse olfactory sensory neurons is determined from soft X-ray tomography reconstructions of five nuclei. We show that there is a division of the nuclear space into regions of low-density euchromatin and high-density heterochromatin. Volume exclusion experienced by a diffusing protein caused by this varying density of chromatin is modeled by a repulsive potential. The value of the potential at a given point in space is chosen to be proportional to the density of chromatin at that location. The constant of proportionality, called the volume exclusivity, provides a model parameter that determines the strength of volume exclusion. Numerical simulations demonstrate that the mean time for a protein to locate a binding site localized in euchromatin is minimized for a finite, nonzero volume exclusivity. For binding sites in heterochromatin, the mean time is minimized when the volume exclusivity is zero (the protein experiences no volume exclusion). An analytical theory is developed to explain these results. The theory suggests that for binding sites in euchromatin there is an optimal level of volume exclusivity that balances a reduction in the volume searched in finding the binding site, with the height of effective potential barriers the protein must cross during the search process.