Optical fibers are an excellent transmission medium for light and underpin the infrastructure of the Internet, but generally after fabrication their optical properties cannot be easily modified. Here, we explore the concept of nanomechanical optical fibers where, in addition to the fiber transmission capability, the internal core structure of the fiber can also be controlled through sub-micron mechanical movements. The nanomechanical functionality of such fibers is demonstrated in the form of dual core optical fibers, in which the cores are independently suspended within the fiber. The movement-based optical change is large compared with traditional electro-optical effects and we show that optical switching of light from one core to the other is achieved through moving one core by just 8 nm.