Gigantism results when one lineage within a clade evolves extremely large body size relative to its small-bodied ancestors, a common phenomenon in animals. Theory predicts that the evolution of giants should be constrained by two tradeoffs. First, because body size is negatively correlated with population size, purifying selection is expected to be less efficient in species of large body size, leading to increased mutational load. Second, gigantism is achieved through generating a higher number of cells along with higher rates of cell proliferation, thus increasing the likelihood of cancer. To explore the genetic basis of gigantism in rodents and uncover genomic signatures of gigantism-related tradeoffs, we assembled a draft genome of the capybara (Hydrochoerus hydrochaeris), the world's largest living rodent. We found that the genome-wide ratio of non-synonymous to synonymous mutations (omega) is elevated in the capybara relative to other rodents, likely caused by a generation-time effect and consistent with a nearly-neutral model of molecular evolution. A genome-wide scan for adaptive protein evolution in the capybara highlighted several genes controlling post-natal bone growth regulation and musculoskeletal development, which are relevant to anatomical and developmental modifications for an increase in overall body size. Capybara-specific gene-family expansions included a putative novel anticancer adaptation that involves T cell-mediated tumor suppression, offering a potential resolution to the increased cancer risk in this lineage. Our comparative genomic results uncovered the signature of an intragenomic conflict where the evolution of gigantism in the capybara involved selection on genes and pathways that are directly linked to cancer.