The coupling of cell-free massive MIMO (CF-mMIMO) with Mobile Edge Computing (MEC) is investigated in this paper. A MEC-enabled CF-mMIMO architecture implementing a distributed user-centric approach both from the radio and the computational resource allocation perspective is proposed. A multi-objective optimization problem (MOOP) for the joint allocation of radio and remote computational resources is formulated, aimed at striking an optimal balance between total uplink power minimization and sum spectral efficiency maximization, under resource budget and latency constraints. In order to solve such a challenging non-convex problem, we convert the MOOP to an equivalent single-objective optimization problem (SOOP) through the weighted sum method and propose an iterative algorithm based on alternating optimization and sequential convex programming, along with an alternative heuristic resource allocation for distributed networks. Finally, we provide a detailed performance comparison between the proposed MEC-enabled CF-mMIMO architecture with its co-located counterpart, and its small-cell implementation. Numerical results reveal the effectiveness of the proposed resource allocation scheme, under different access point selection strategies, and the natural suitability of CF-mMIMO in supporting computation-offloading applications with benefits over users' transmit power and energy consumption, the effective latency experienced, and the computation offloading efficiency.