Deep Neural Networks (DNNs) are a promising tool for Global Navigation Satellite System (GNSS) positioning in the presence of multipath and non-line-of-sight errors, owing to their ability to model complex errors using data. However, developing a DNN for GNSS positioning presents various challenges, such as 1) poor numerical conditioning caused by large variations in measurements and position values across the globe, 2) varying number and order within the set of measurements due to changing satellite visibility, and 3) overfitting to available data. In this work, we address the aforementioned challenges and propose an approach for GNSS positioning by applying DNN-based corrections to an initial position guess. Our DNN learns to output the position correction using the set of pseudorange residuals and satellite line-of-sight vectors as inputs. The limited variation in these input and output values improves the numerical conditioning for our DNN. We design our DNN architecture to combine information from the available GNSS measurements, which vary both in number and order, by leveraging recent advancements in set-based deep learning methods. Furthermore, we present a data augmentation strategy for reducing overfitting in the DNN by randomizing the initial position guesses. We first perform simulations and show an improvement in the initial positioning error when our DNN-based corrections are applied. After this, we demonstrate that our approach outperforms a WLS baseline on real-world data. Our implementation is available at github.com/Stanford-NavLab/deep_gnss.