To be clear, there are many ways to calculate absolute or relative molecular orbital/state energies, some of them completely theoretical, some of them semi-empirical. Anything other than a 1 electron system is by nature an approximation rather than calculation of an exact value, and the vast majority of reasonably good calculations require computers because they are mathematically intensive. Some types of systems are far easier than others, but by and large we can do a reasonably good job of calculating molecular state energies, and also molecular structures, even with reasonably simple approximations.
All that said, no calculation will ever replace experimental data. Calculations are most useful in cases were experiments are just not easy to pull off (e.g., states that are spectroscopically forbidden, nonequilibrium structures, transition states or other quasi-stable states, and so forth).