This work proposes the ROMES (Reduced-Order-Model Error Surrogates) method for statistical closure modeling of reduced-order models for stationary systems. The method constructs statistical models to characterize the error incurred by reduced-order models, enabling accurate uncertainty quantification even when the reduced-order model itself is approximate. We consider both linear and nonlinear stationary systems and demonstrate the method's ability to provide accurate error estimates across a range of parameters.