Massive multiple-input multiple-output (MIMO) communications are the focus of considerable interest in recent years. While theoretical gains of such massive MIMO have been established, implementing MIMO systems with large-scale antenna arrays in practice is challenging. Among the practical difficulties associated with massive MIMO implementations are increased cost, power consumption, and physical size. In this work we study the implementation of massive MIMO antenna arrays using dynamic metasurface antennas (DMAs), an emerging technology which inherently handles the aforementioned challenges. DMAs realize planar large-scale arrays of tunable antenna elements, and can adaptively incorporate compression and analog combining in the physical antenna structure, thus reducing cost and power consumption. We first propose a mathematical model for massive MIMO systems with DMAs and discuss their constraints compared to ideal antenna arrays. Then, we characterize the fundamental limits of the resulting systems, and propose an algorithm for designing practical DMAs to approach these limits. Our numerical results indicate that the performance of practical DMA-based massive MIMO systems is comparable with ideal antenna arrays.