Accumulation of gaseous methane can generate seabed domes in soft cohesive sediments. Such structures can, in turn, lead to seafloor instability, act as precursors to pockmark formation, and arguably pose a threat to seafloor drilling if they contain significantly overpressured gas. Future melting of gas hydrates within the seabed, due to global warming, will likely lead to a significant long-term release of methane, which could potentially produce a new and abundant generation of gas domes and associated pockmarks. Despite their geological and practical significance, our understanding of gas-dome formation in marine sediments has been limited to observations and qualitative analyses. To provide a quantitative understanding, we conducted small-scale laboratory doming experiments. We found that thin layers of clayey sediment behave elastically over the range of deformations needed to create seabed domes. The observed behavior is well described by elastic thin-plate mechanics, from which it is possible to predict the gas pressure required to create natural domes. Our results suggest that observed shallow dome geometries require surprisingly small overpressures to form; however, large overpressures can build under increasingly thicker and stiffer layers of sediment.