The observation of high mitochondrial mutation rates in human pedigrees has led to the question of how such an asexual genetic system can survive the accumulation of slightly deleterious mutations caused by Muller's ratchet. I define a null model to quantify in unprecedented detail the threat from extinction caused by Muller's ratchet. This model is general enough to explore the biological significance of Muller's ratchet in various species where its operation has been suspected. For increased precision over a wide range of parameter space I employ individual-based simulations run by evolution@home, the first global computing system for evolutionary biology. After compiling realistic values for the key parameters in human mitochondrial DNA (mtDNA) I find that a surprisingly large range of biologically realistic parameter combinations would lead to the extinction of the human line over a period of 20 million years - if accepted wisdom about mtDNA and Muller's ratchet is correct. The resulting genomic decay paradox complements a similar threat from extinction due to mutation accumulation in nuclear DNA and suggests evaluation of unconventional explanations for long-term persistence. A substantial list of potential solutions is given, including compensatory back mutations, mutation rate heterogeneity and occasional recombination in mtDNA. Future work will have to explore which of these actually solves the paradox. Nonetheless, the results presented here provide yet another reason to minimize anthropogenic increase of mutation rates.