Distributed storage systems (DSSs) have gained a lot of interest recently, thanks to their robustness and scalability compared to single-device storage. Majority of the related research has exclusively concerned the network layer. At the same time, the number of users of, e.g., peer-to-peer (p2p) and device-to-device (d2d) networks as well as proximity based services is growing rapidly, and the mobility of users is considered more and more important. This motivates, in contrast to the existing literature, the study of the physical layer functionality of wireless distributed storage systems. In this paper, we take the first step towards protecting the storage repair transmissions from physical layer errors when the transmission takes place over a fading channel. To this end, we introduce the notion of a space-time storage code, drawing together the aspects of network layer and physical layer functionality and resulting in cross-layer robustness. It is also pointed out that existing space-time codes are too complex to be utilized in storage networks when the number of helpers involved is larger than the number of receive antennas at the newcomer or data collector, hence creating a call for less complex transmission protocols.