Soft robots use compliance to interact with and navigate complex environments. High-resolution shape sensing is required to leverage this mechanical advantage by enabling feedback control and environmental mapping. Fiber optic sensing is a promising approach for shape estimation in soft robots. However, commercial fiber-optic shape sensors are prohibitively expensive for many applications in research and development due to their extensive data acquisition apparatus. Moreover, low-cost solutions that have been developed in research settings lack the sensing range of these commercial sensors and can only estimate discrete bending angles or a single radius of curvature. We present a fiber optic sensor that enables multi-curve shape sensing with sub-percent prediction error using an off-the-shelf camera and multiple inexpensive PMMA optical fibers for sensing. This paper introduces a sensor fabrication method based on laser-engraving optical fibers and a data-processing pipeline based on conventional image processing and machine learning tools for interpreting sensor output. We furthermore provide an analytical model to inform sensor design based on the required spatial resolution. Finally, we report empirical findings from several prototypes to highlight the sensor's unique capabilities.