Compression and efficient storage of neural network (NN) parameters is critical for applications that run on resource-constrained devices. Although NN model compression has made significant progress, there has been considerably less investigation in the actual physical storage of NN parameters. Conventionally, model compression and physical storage are decoupled, as digital storage media with error correcting codes (ECCs) provide robust error-free storage. This decoupled approach is inefficient, as it forces the storage to treat each bit of the compressed model equally, and to dedicate the same amount of resources to each bit. We propose a radically different approach that: (i) employs analog memories to maximize the capacity of each memory cell, and (ii) jointly optimizes model compression and physical storage to maximize memory utility. We investigate the challenges of analog storage by studying model storage on phase change memory (PCM) arrays and develop a variety of robust coding strategies for NN model storage. We demonstrate the efficacy of our approach on MNIST, CIFAR-10 and ImageNet datasets for both existing and novel compression methods. Compared to conventional error-free digital storage, our method has the potential to reduce the memory size by one order of magnitude, without significantly compromising the stored model's accuracy.