We study the multifractal analysis of the pointwise dimension for equilibrium measures on countable Markov shifts. The main difficulty is that the space is not compact. In order to overcome this, we use an approximation argument based on the theory of convergence of Fenchel pairs developed by Wijsman. The results of Pesin and Weiss on multifractal analysis for compact spaces are used as well. We also prove a Bowen formula for countable Markov shifts. It turns out that, in this setting, this formula provides the Hausdorff dimension of the set of recurrent points.