One of the key challenges in sensing networks is the extraction of information by fusing data from a multitude of possibly unreliable sensors. Robust sensing, viewed here as the simultaneous recovery of the wanted information-bearing signal vector together with the subset of (un)reliable sensors, is a problem whose optimum solution incurs combinatorial complexity. The present paper relaxes this problem to its closest convex approximation that turns out to yield a vector-generalization of Huber's scalar criterion for robust linear regression. The novel generalization is shown equivalent to a second-order cone program (SOCP), and exploits the block-sparsity inherent to a suitable model of the residuals. A computationally efficient solver is developed using a block-coordinate descent algorithm, and is tested with simulations.