To fully benefit from remotely sensed observations of the terrestrial water cycle, bias and random errors in these datasets need to be quantified. This paper presents a Bayesian hierarchical model that fuses monthly water balance data and estimates the corresponding data errors and error-corrected water balance components (precipitation, evaporation, river discharge, and water storage). The model combines monthly basin-scale water balance constraints with probabilistic data error models for each water balance variable. Each data error model includes parameters that are in turn treated as unknown random variables to reflect uncertainty in the errors. Errors in precipitation and evaporation data are parameterized as a function of multiple data sources, while errors in GRACE storage observations are described by a noisy sine wave model with parameters controlling phase, amplitude and randomness of the sine wave. Error parameters and water balance variables are estimated using a combination of Markov Chain Monte Carlo sampling and iterative smoothing. Application to semi-arid river basins in Iran yields (i) significant reductions in evaporation uncertainty during water-stressed summers, (ii) basin-specific timing and amplitude corrections of the GRACE water storage dynamics, and (iii) posterior water balance estimates with average standard errors of 4-12 mm/month for water storage, 3.5-7 mm/month for precipitation, 2-6 mm/month for evaporation, and 0-2 mm/month for river discharge. The approach is readily extended to other datasets and other (gauged) basins around the world, possibly using customized data error models. The resulting error-filtered and bias-corrected water balance estimates can be used to evaluate hydrological models.