The main result of the paper consists of a fine analysis of the size of the set \(G_\mu\) of generic points (i.e., satisfying the Birkhoff ergodic theorem for all continuous functions) of an invariant measure \(\mu\) defined on a symbolic space. The symbolic space is endowed with a metric defined via a Gibbs measure \(\mu_\phi\) associated to a potential \(\phi\). It is proved that, for any gauging function \(g\) with the doubling property, both the Hausdorff and packing \(g\)-measures of \(G_\mu\) --for \(\mu\not =\mu_\phi\)-- are either zero or infinity. The result characterizes completely which is the case for each Hausdorff (packing) \(g\)-measure, depending whether the upper (lower) limit of \({\log g(t) /\log t}\) is above or below \(s:={h_\mu\over P_\phi-\int \phi d\mu}\) -- here \(h_\mu\) is the measure-theoretic entropy and \(P_\phi\) is the topological pressure of \(\phi\). As a consequence, \(G_\mu\) has dimensions given by \(s\) and infinite \(s\)-dimensional Hausdorff and packing measures. The dimension formula above arises in different geometric and dynamical settings considered in [\textit{Y. B. Pesin}, ``Dimension theory in dynamical systems'' (1997; Zbl 0895.58033)]. As an application, it is shown that level sets of Birkhoff averages in the symbolic space have infinite Hausdorff measure in their dimension. Also, the main results are translated to the context of self-similar geometric constructions -- seen as projections of the symbolic space -- by linking Hausdorff measures and dimensions in the self-similar space with their counterparts in the symbol space. (This link was derived in a more general setting in [\textit{J.-M. Rey}, Proc. Am. Math. Soc. 128, No. 2, 561--572 (2000; Zbl 0938.28002)].)