We present aperture synthesis images of the CO (1-0) line emission in five central galaxies in cooling flow clusters using the Owens Valley Millimeter Array. Three of the five sources are significantly resolved, but the majority of the emission is from a compact (<20 kpc) region centered on the central galaxy. These results are consistent with the newly emerging view that cooling flows deposit much less gas over a shorter period in a smaller volume than previously thought. The size constraints derived imply that the molecular gas has a large (> 10(22) cm(-2)) column density. We review the implications of these results and the prospects for observations in the near future.