We report the results of a pilot program to use the Magellan/M2FS spectrograph to survey the galactic populations and internal kinematics of galaxy clusters. For this initial study, we present spectroscopic measurements for 223 quiescent galaxies observed along the line of sight of the galaxy cluster Abell 267 (z~0.23). We develop a Bayesian method for modeling the integrated light from each galaxy as a simple stellar population, with free parameters that specify the redshift (v_los_/c) and characteristic age, metallicity ([Fe/H]), alpha-abundance ([{alpha}/Fe]), and internal velocity dispersion ({sigma}_int_) for individual galaxies. Parameter estimates derived from our 1.5hr observation of A267 have median random errors of {sigma}_Vlos_=20km/s, {sigma}_Age_=1.2Gyr, {sigma}_[Fe/H]_=0.11dex, {sigma}_[{alpha}/Fe]_=0.07dex, and {sigma}_{sigma}int_=20km/s. In a companion paper, we use these results to model the structure and internal kinematics of A267.