Designing capacity-achieving coding schemes for the band-limited additive colored Gaussian noise (ACGN) channel has been and is still a challenge. In this paper, the capacity of the band-limited ACGN channel is studied from a fundamental algorithmic point of view by addressing the question of whether or not the capacity can be algorithmically computed. To this aim, the concept of Turing machines is used, which provides fundamental performance limits of digital computers. It is shown that there are band-limited ACGN channels having computable continuous spectral densities whose capacity are non-computable numbers. Moreover, it is demonstrated that for those channels, it is impossible to find computable sequences of asymptotically sharp upper bounds for their capacities. Furthermore, the implications of the non-computability of the ACGN channel capacity in information theory and coding are discussed, particularly regarding the impossibility of computing achievable rates in the finite blocklength regime and the challenges of finding universal algorithms that compute capacity-achieving power spectral densities for the ACGN channel.