We present the first statistical study on the intrinsic three-dimensional (3D) shape of a sample of 83 galactic bars extracted from the CALIFA survey. We use the galaXYZ code to derive the bar intrinsic shape with a statistical approach. The method uses only the geometric information (ellipticities and position angles) of bars and discs obtained from a multicomponent photometric decomposition of the galaxy surface-brightness distributions. We find that bars are predominantly prolate-triaxial ellipsoids (68 per cent), with a small fraction of oblate-triaxial ellipsoids (32 per cent). The typical flattening (intrinsic C/A semiaxis ratio) of the bars in our sample is 0.34, which matches well the typical intrinsic flattening of stellar discs at these galaxy masses. We demonstrate that, for prolate-triaxial bars, the intrinsic shape of bars depends on the galaxy Hubble type and stellar mass (bars in massive S0 galaxies are thicker and more circular than those in less massive spirals). The bar intrinsic shape correlates with bulge, disc, and bar parameters. In particular with the bulge-to-total (B/T) luminosity ratio, disc g - r colour, and central surface brightness of the bar, confirming the tight link between bars and their host galaxies. Combining the probability distributions of the intrinsic shape of bulges and bars in our sample, we show that 52 per cent (16 per cent) of bulges are thicker (flatter) than the surrounding bar at 1σ level. We suggest that these percentages might be representative of the fraction of classical and disc-like bulges in our sample, respectively.