We describe an optomechanical device consisting of a fiber-based optical cavity containing a silicon nitride membrane. In comparison with typical free-space cavities, the fiber-cavity's small mode size (10 μm waist, 80 μm length) allows the use of smaller, lighter membranes and increases the cavity-membrane linear coupling to 3 GHz/nm and the quadratic coupling to 20 GHz/nm2. This device is also intrinsically fiber-coupled and uses glass ferrules for passive alignment. These improvements will greatly simplify the use of optomechanical systems, particularly in cryogenic settings. At room temperature, we expect these devices to be able to detect the shot noise of radiation pressure.