Abstract: Detecting primordial fluctuations from the cosmic dark ages requires extremely large low-frequency radio telescope arrays deployed on the far side of the Moon. The antenna of such an array must be lightweight, easily storable and transportable, deployable on a large scale, durable, and capable of good electrical performance. A membrane antenna is an excellent candidate to meet these criteria. We study the design of a low-frequency membrane antenna for a lunar-based low-frequency (<30 MHz) radio telescope constructed from polyimide film widely used in aerospace applications, owing to its excellent dielectric properties and high stability as a substrate material. We first design and optimize an antenna in free space through dipole deformation and coupling principles, then simulate an antenna on the lunar surface with a simple lunar soil model, yielding an efficiency greater than 90% in the range of 12–19 MHz and greater than 10% in the range of 5–35 MHz. The antenna inherits the omni-directional radiation pattern of a simple dipole antenna in the 5–30 MHz frequency band, giving a large field of view and allowing detection of the 21 cm global signal when used alone. A demonstration prototype is constructed, and its measured electrical property is found to be consistent with simulated results using |S₁₁| measurements. This membrane antenna can potentially fulfill the requirements of a lunar low-frequency array, establishing a solid technical foundation for future large-scale arrays for exploring the cosmic dark ages.