The paper presents a novel three-dimensional quasi-continuous shape sensor based on an FBG array inscribed by femtosecond laser pulses into a 7-core optical fiber with a polyimide protective coating. The measured bending sensitivity of individual FBGs ranges from 0.046 nm/m−1 to 0.049 nm/m−1. It is shown that the sensor allows for reconstructing 2- and 3-dimensional shapes with high accuracy. Due to the high value of the core aperture and individual calibration of each FBG we were able to measure the smallest reported bending radii down to 2.6 mm with a record accuracy of ∼1%. Moreover, we investigate the magnitude of the errors of curves reconstruction and errors associated with measurement of curvature radii in the range from 2.6 to 500 mm. The main factors affecting the accuracy of measurements are also discussed. The temperature resistance of both the inscribed FBG structures and of the protective coating, along with the high mechanical strength of the polyimide, makes it possible to use the sensor in harsh environments or in medical and composite material applications.