The transition from prokaryotic lateral gene transfer to eukaryotic meiotic sex is poorly understood. Phylogenetic evidence suggests that it was tightly linked to eukaryogenesis, which involved an unprecedented rise in both genome size and the density of genetic repeats. Expansion of genome size raised the severity of Muller's ratchet, whilst limiting the effectiveness of lateral gene transfer (LGT) at purging deleterious mutations. In principle, an increase in recombination length combined with higher rates of LGT could solve this problem. Here we show using a computational model that this solution fails in the presence of genetic repeats prevalent in early eukaryotes. The model demonstrates that dispersed repeat sequences allow ectopic recombination, which leads to the loss of genetic information and curtails the capacity of LGT to prevent mutation accumulation. Increasing recombination length in the presence of repeat sequences exacerbates the problem. Mutational decay can only be resisted with homology along extended sequences of DNA. We conclude that the transition to homologous pairing along linear chromosomes was a key innovation in meiotic sex, which was instrumental in the expansion of eukaryotic genomes and morphological complexity.