In particular, a system usually cannot prove itself consistent. However, we have developed systems expressly for the purpose of being as consistent as our intuition for logic could allow it to be. I'm willing to trust Metamath [0] based on the formalism that they've chosen, which is a dead-simple term-rewriting formalism that is suitable for building pretty much any mathematical structure, given enough time and effort.
A better question would be whether, in a theoretical sense, fault-free algorithms even exist. There are similar questions, like whether one-way functions exist, where we have lots of evidence to suggest that the answer is "yes" but we have yet to prove it.