Vector unparticle couplings to standard model fields produce repulsive corrections to gravity. From a general relativistic perspective, this leads to an effective Reissner-Nordstrom-like metric whose "charge" is a function of the unparticle coupling constant λ, and therefore can admit naked singularities. Requiring the system to respect cosmic censorship provides a new method of constraining the value of λ. These limits are extremely loose for stellar-mass black holes, but commensurate with existing bounds for primordial black holes. In the case of theoretical low-mass black holes, the bounds on λ are much stricter than those derived from astrophysical and accelerator phenomenology. Additional constraints on the lower limit of λ are used to estimate the mass of the smallest possible black hole MminBH that can be formed in the unparticle framework, as a function of the unparticle parameters (LambdaU;MU; dU; dBZ).