Given full or partial information about a collection of points that lie close to a union of several subspaces, subspace clustering refers to the process of clustering the points according to their subspace and identifying the subspaces. One popular approach, sparse subspace clustering (SSC), represents each sample as a weighted combination of the other samples, with weights of minimal $\ell_1$ norm, and then uses those learned weights to cluster the samples. SSC is stable in settings where each sample is contaminated by a relatively small amount of noise. However, when there is a significant amount of additive noise, or a considerable number of entries are missing, theoretical guarantees are scarce. In this paper, we study a robust variant of SSC and establish clustering guarantees in the presence of corrupted or missing data. We give explicit bounds on amount of noise and missing data that the algorithm can tolerate, both in deterministic settings and in a random generative model. Notably, our approach provides guarantees for higher tolerance to noise and missing data than existing analyses for this method. By design, the results hold even when we do not know the locations of the missing data; e.g., as in presence-only data.