Earth's temperature variability can be partitioned into internal and externally forced components. Yet, underlying mechanisms and their relative contributions remain insufficiently understood, especially on decadal to centennial timescales. Important reasons for this are difficulties in isolating internal and externally forced variability. Here, we provide a physically motivated emulation of global mean surface temperature (GMST) variability, which allows for the separation of internal and external variations. To this end, we introduce the "ClimBayes" software package, which infers climate parameters from a stochastic energy balance model (EBM) with a Bayesian approach. We apply our method to GMST data from temperature observations and 20 last millennium simulations from climate models of intermediate to high complexity. This yields the best estimates of the EBM's forced and forced + internal response, which we refer to as emulated variability. The timescale-dependent variance is obtained from spectral analysis. In particular, we contrast the emulated forced and forced + internal variance on interannual to centennial timescales with that of the GMST target. Our findings show that a stochastic EBM closely approximates the power spectrum and timescale-dependent variance of GMST as simulated by modern climate models. Small deviations at interannual timescales can be attributed to the simplified representation of internal variability and, in particular, the absence of (pseudo-)oscillatory modes in the stochastic EBM. Altogether, we demonstrate the potential of combining Bayesian inference with conceptual climate models to emulate statistics of climate variables across timescales.