We measure the organization and substructure of HT29 epithelial cells in a monolayer using angle-resolved low-coherence interferometry. This new technique probes cellular structure by measuring scattered light, as in flow cytometry, but offers an advantage in that the structure can be examined in situ, avoiding the need to disrupt the cell monolayer. We determine the size distribution of the cell nuclei by fitting measured light-scattering spectra to the predictions of Mie theory. In addition, we obtain information about the cellular organization and substructure by examining the spatial correlations within the monolayer. A remarkable finding is that the spatial correlations over small length scales take the form of an inverse power law, indicating the fractal nature of the packing of the subcellular structures. We also identify spatial correlations on a scale large compared with the size of a cell, indicating an overlying order within the monolayer.