While learning from synthetic training data has recently gained an increased attention, in real-world robotic applications, there are still performance deficiencies due to the so-called Sim-to-Real gap. In practice, this gap is hard to resolve with only synthetic data. Therefore, we focus on an efficient acquisition of real data within a Sim-to-Real learning pipeline. Concretely, we employ deep Bayesian active learning to minimize manual annotation efforts and devise an autonomous learning paradigm to select the data that is considered useful for the human expert to annotate. To achieve this, a Bayesian Neural Network (BNN) object detector providing reliable uncertainty estimates is adapted to infer the informativeness of the unlabeled data. Furthermore, to cope with mis-alignments of the label distribution in uncertainty-based sampling, we develop an effective randomized sampling strategy that performs favorably compared to other complex alternatives. In our experiments on object classification and detection, we show benefits of our approach and provide evidence that labeling efforts can be reduced significantly. Finally, we demonstrate the practical effectiveness of this idea in a grasping task on an assistive robot.