This paper shows that the epsilon entropy of any mean-continuous Gaussian process on $L_2\lbrack 0, 1 \rbrack$ is finite for all positive $\epsilon$. The epsilon entropy of such a process is defined as the infimum of the entropies of all partitions of $L_2\lbrack 0, 1 \rbrack$ by measurable sets of diameter at most $\epsilon$, where the probability measure on $L_2$ is the one induced by the process. Fairly tight upper and lower bounds are found as $\epsilon \rightarrow 0$ for the epsilon entropy in terms of the eigenvalues of the process.