Self-adjoint operators and their spectra play a crucial role in analysis and physics. For instance, in quantum physics self-adjoint operators are used to describe measurements and the spectrum represents the set of possible measurement results. Therefore, it is a natural question whether the spectrum of a self-adjoint operator can be computed from a description of the operator. We prove that given a "program" of the operator one can obtain positive information on the spectrum as a compact set in the sense that a dense subset of the spectrum can be enumerated (or equivalently: its distance function can be computed from above) and a bound on the set can be computed. This generalizes some non-uniform results obtained by Pour-El and Richards, which imply that the spectrum of any computable self-adjoint operator is a recursively enumerable compact set. Additionally, we show that the spectrum of compact self-adjoint operators can even be computed in the sense that also negative information is available (i.e. the distance function can be fully computed). Finally, we also discuss computability properties of the resolvent map.