A thin and stretchable polymer layer can be fabricated over large areas with high uniformity using a vacuum-deposition method and used as the gate dielectric in stretchy carbon-nanotube-based transistors and circuits that can function at 40% strain. Despite recent advances in materials and fabrication technologies, the development of intrinsically stretchable electronic devices with large-area uniformity, low power consumption and performance comparable with conventional rigid devices remains challenging. A key limitation is the absence of an elastic dielectric material that can be thin and uniform over large areas as well as offer robust insulating properties and high mechanical and chemical stability. Here we show that a vacuum-deposited elastic polymer layer can be used as the gate dielectric in stretchy field-effect transistors with carbon nanotube channels and microcracked gold electrodes. The polymer dielectric layer has high insulation properties (at a thickness below 200 nm), high stability and large-area uniformity. An 8-inch wafer array of the stretchable carbon nanotube transistors exhibits good uniformity in their electrical performance and can maintain their performance after 1,000 stretching cycles at 40% strain. We use the transistors to construct stretchy inverters and logic gates that can function under applied strains of up to 40%.