We present the results from a study of the spatial distributions of cooling gas and intrinsic X-ray-absorbing material in a sample of nearby, X-ray-bright cooling flow clusters observed with the Position Sensitive Proportional Counter (PSPC) on ROSAT. Our method of analysis employs X-ray colour profiles, formed from ratios of the surface brightness profiles of the clusters in selected energy bands, and an adapted version of the deprojection code of Fabian et al. We show that all of the cooling flow clusters in our sample exhibit significant central concentrations of cooling gas. At larger radii the clusters appear approximately isothermal. In detail, the spatial distributions and emissivity of the cooling material are shown to be in excellent agreement with the predictions from the deprojection code, and can be used to constrain the ages of the cooling flows. The X-ray colour profiles also indicate substantial levels of intrinsic X-ray absorption in the clusters. The intrinsic absorption increases with decreasing radius, and is confined to the regions occupied by the cooling flows. We explore a range of likely spatial distributions for the absorbing gas and discuss the complexities involved in the measurement of column densities from X-ray data. We show that the application of simple spectral models, in which the intrinsic absorber is treated as a uniform foreground screen, will naturally lead to significant underestimates of the true amounts of absorption. Comparison of our results with previously reported observations made with the Einstein-Observatory Solid State Spectrometer shows reasonable agreement, but requires that the absorbing material only partially covers the X-ray-emitting regions. The mass of absorbing gas in the central (30 arcsec radius) regions of the clusters calculated under the assumption of solar metallicity in the absorbing material can be accumulated by the cooling flows on time-scales of a few 108 yr, which are much less than the ages of the flows. This implies that most of the material deposited by the cooling flows in these regions cannot remain in X-ray-absorbing gas. The results presented in this paper provide strong support for the standard model of inhomogeneous cooling flows in clusters of galaxies.