Extreme optical properties can be realized by the strong resonant response of metamaterials consisting of subwavelength-scale metallic resonators. However, highly dispersive optical properties resulting from strong resonances have impeded the broadband operation required for frequency-independent optical components or devices. Here we demonstrate that strong, flat broadband optical activity with high transparency can be obtained with meshed helical metamaterials in which metallic helical structures are networked and arranged to have fourfold rotational symmetry around the propagation axis. This nondispersive optical activity originates from the Drude-like response as well as the fourfold rotational symmetry of the meshed helical metamaterials. The theoretical concept is validated in a microwave experiment in which flat broadband optical activity with a designed magnitude of 45 degrees per layer of metamaterial is measured. The broadband capabilities of chiral metamaterials may provide opportunities in the design of various broadband optical systems and applications.