The stochastic calibration of low-cost and consumer grade inertial sensors has recently become very important due to their wide-spread utilization in a multitude of mass-market applications like smartphone and drone navigation. The reason behind this is because if accurate stochastic modeling about the inertial sensor noise is obtained, then the estimation quality of the navigation solution may improve significantly. Generally, the mainstream methods for stochastic calibration consider only a single signal, collected under static conditions, to infer that knowledge. However, it has been observed that even though the stochastic model structure that characterizes each (static) calibration signal remains the same, its parameter values vary from one replicate to another. Even though techniques have been recently proposed to address this in a statistically efficient way, a very important factor has been neglected, namely the influence of outliers on the estimation process. In this paper, a robust multi-signal framework for the stochastic modeling of inertial sensor errors is proposed, which contains two layers of robustness: one that reduces the influence of outliers in each observed signal (data corruption) and one that safeguards the estimation process from the collection of calibration signal replicates with notably different stochastic behaviour compared to the majority (sample contamination). Furthermore, two estimators are defined from this framework, with each encompassing either one or both layers of robustness, and their efficiency in different data contamination scenarios is assessed in a simulation setting. Finally, real data collected from a consumer-grade MEMS-based device are used within a navigation simulator to evaluate the relationship between the quality of the stochastic models obtained by the two robust estimators in different data collection scenarios and the navigation solution stability.