Calculation of band gap

As was already pointed out in the answer of @CGS the basis for calculations on the band structure typically is Density functional theory (DFT). People interpret the eigenvalues of the related Kohn-Sham states to be forming the band structure. On a semiquantitative level that is acceptable and the band structure obtained in this way can provide detailed insights into the electronic structure of the material. But in a stringent evaluation of this approach these Kohn-Sham eigenvalues have no strict physical interpretation in that way.

Evaluating the band gap in this way typically underestimates the real gap, in comparison to Hartree-Fock calculations wich typically overestimate the gap. The underestimation of the gap can go as far as predicting a material to be a metal while it actually is an insulator. This can often be observed for Mott insulators where the band gap depends on the correct treatment of strong electron correlations: Exchange and correlation interactions are approximated in practical realizations of DFT. To reduce this problem there are many approximations to the "exchange-correlation (XC) functional". For simple calculations one uses the local density approximation (LDA) or a generalized gradient approximation (GGA). To get better estimates on the band gap one can also employ a combination of LDA with a Hubbard U term in LDA+U or use Hybrid functionals like the already mentioned HSE functional. The latter mix LDA, GGA, and Hartree-Fock-like exchange.

But even if one constructs the exact exchange-correlation functional in this way the Kohn-Sham eigenvalues would not predict the band gap. Strictly speaking this quantity is just not predicted in this way.

A more direct simulation of the physics playing a role for the band gap is the GW approximation to many-body perturbation theory. To calculate such quantities the GW approximation is typically used on top of DFT results. You mentioned spectroscopic methods to measure the band gap. The GW approximation is strongly related to such methods. See, e.g., this paper: https://doi.org/10.3389/fchem.2019.00377

Finally, one should mention that the GW approximation is still an approximation. There are many effects playing a role in the determination of the size of the band gap. For example, excitonic effects tend to narrow the gap. Such excitonic effects are not covered by the GW approximation. Here one has to employ the Bethe-Salpeter equation. Other effects may require other additions to the model.

Essentially this means that one has to understand the relevant physics in a material to choose an adequate simulation mehod and obtain a good prediction of the band gap without wasting too much computation time.