A novel kind of nanostructured optical fiber, displaying an extremely high and optically broadband optomechanical nonlinearity, is presented. It comprises two closely spaced ultrathin glass membranes (webs) suspended in air and attached to the inner walls of a glass fiber capillary. Light guided in this dual-web structure can exert attractive or repulsive pressure on the webs, causing them to be pushed together or pulled apart. The elastic deflection of the webs is, in turn, coupled to the electromagnetic field distribution and results in a change in the effective refractive index within the fiber. Employing a pump-probe technique in an interferometric setup, optomechanically induced refractive index changes more than 10(4) times larger than the Kerr effect are detected. Theoretical estimates of the optomechanical nonlinearity agree well with the experimental results. The dual-web fiber combines the sensitivity of a microoptomechanical device with the versatility of an optical fiber and could trigger new developments in the fields of nonlinear optics, optical metrology, and sensing.