In this paper we consider the problem of recovering a high dimensional data matrix from a set of incomplete and noisy linear measurements. We introduce a new model that can efficiently restrict the degrees of freedom of the problem and is generic enough to find a lot of applications, for instance in multichannel signal compressed sensing (e.g. sensor networks, hyperspectral imaging) and compressive sparse principal component analysis (s-PCA). We assume data matrices have a simultaneous low-rank and joint sparse structure, and we propose a novel approach for efficient compressed sensing (CS) of such data. Our CS recovery approach is based on a convex minimization problem that incorporates this restrictive structure by jointly regularizing the solutions with their nuclear (trace) norm and l2/l1 mixed norm. Our theoretical analysis uses a new notion of restricted isometry property (RIP) and shows that, for sampling schemes satisfying RIP, our approach can stably recover all low-rank and joint-sparse matrices. For a certain class of random sampling schemes satisfying a particular concentration bound (e.g. the subgaussian ensembles) we derive a lower bound on the number of CS measurements indicating the near-optimality of our recovery approach as well as a significant enhancement compared to the state-of-the-art. We introduce an iterative algorithm based on proximal calculus in order to solve the joint nuclear and l2/l1 norms minimization problem and, finally, we illustrate the empirical recovery phase transition of this approach by series of numerical experiments.