A framework that incorporates channel modeling, position estimation, and error analysis methods is developed for network-based indoor positioning with radio signal strength (RSS) measurements of WiFi access points (APs). To construct an accurate channel model with RSS measurements severely influenced by propagation attenuations, multipath reflections, and shadowing effects, a novel sparse Bayesian learning algorithm is developed to model the radio power map (RPM) in indoor space. Based on the proposed RPM model, a 2-stage positioning method is further developed. In the first stage for coarse positioning, the location is determined up to a room-scale indoor space. Then, in the second stage for fine positioning, the RPMs of the given indoor space are used for location estimation in the space with a Bayesian estimator. The mean squared positioning errors are verified with the Bayesian Cramer-Rao lower bound. Extensive experiments show that the average positioning error of the proposed RPM-based approach is 1.98 meters which achieve 22 percent improvements over the state-of-the-art RSS-based indoor positioning methods. More importantly, the proposed modeling and positioning method can effectively exploit the spatial relationship in the RSS samples to improve positioning accuracy.