We present Chandra/ACIS-S subarray observations of the quiescent neutron star (NS) low-mass X-ray binaries X7 and X5 in the globular cluster 47 Tuc. The large reduction in photon pile-up compared to previous deep exposures enables a substantial improvement in the spectroscopic determination of the NS radius and mass of these NSs. Modeling the thermal emission from the NS surface with a non-magnetized hydrogen atmosphere and accounting for numerous sources of uncertainties, we obtain for the NS in X7 a radius of R = 11.1(-0.7)(+0.8) km for an assumed stellar mass of M = 1.4 M-circle dot (68% confidence level). We argue, based on astrophysical grounds, that the presence of a He atmosphere is unlikely for this source. Due to the excision of data affected by eclipses and variable absorption, the quiescent low-mass X-ray binary X5 provides less stringent constraints, leading to a radius of R = 9.6(-1.1)(+0.9) km, assuming a hydrogen atmosphere and a mass of M = 1.4 M-circle dot. When combined with all existing spectroscopic radius measurements from other quiescent low-mass X-ray binaries and Type I X-ray bursts, these measurements strongly favor radii in the 9.9-11.2 km range for a similar to 1.5 M-circle dot NS and point to a dense matter equation of state that is somewhat softer than the nucleonic ones that are consistent with laboratory experiments at low densities.