Studying the solar system and especially the Sun relies on the data gathered daily from space missions. These missions are data-intensive and compressing this data to make them efficiently transferable to the ground station is a twofold decision to make. Stronger compression methods, by distorting the data, can increase data throughput at the cost of accuracy which could affect scientific analysis of the data. On the other hand, preserving subtle details in the compressed data requires a high amount of data to be transferred, reducing the desired gains from compression. In this work, we propose a neural network-based lossy compression method to be used in NASA's data-intensive imagery missions. We chose NASA's SDO mission which transmits 1.4 terabytes of data each day as a proof of concept for the proposed algorithm. In this work, we propose an adversarially trained neural network, equipped with local and non-local attention modules to capture both the local and global structure of the image resulting in a better trade-off in rate-distortion (RD) compared to conventional hand-engineered codecs. The RD variational autoencoder used in this work is jointly trained with a channel-dependent entropy model as a shared prior between the analysis and synthesis transforms to make the entropy coding of the latent code more effective. Our neural image compression algorithm outperforms currently-in-use and state-of-the-art codecs such as JPEG and JPEG-2000 in terms of the RD performance when compressing extreme-ultraviolet (EUV) data. As a proof of concept for use of this algorithm in SDO data analysis, we have performed coronal hole (CH) detection using our compressed images, and generated consistent segmentations, even at a compression rate of $\sim0.1$ bits per pixel (compared to 8 bits per pixel on the original data) using EUV data from SDO.