This paper initiates the study of fractional eternal domination in graphs, a natural relaxation of the well-studied eternal domination problem. We study the connections to flows and linear programming in order to obtain results on the complexity of determining the fractional eternal domination number of a graph $G$, which we denote $γ_{\,\textit{f}}^{\infty}(G)$. We study the behaviour of $γ_{\,\textit{f}}^{\infty}(G)$ as it relates to other domination parameters. We also determine bounds on, and in some cases exact values for, $γ_{\,\textit{f}}^{\infty}(G)$ when $G$ is a member of one of a variety of important graph classes, including trees, split graphs, strongly chordal graphs, Kneser graphs, abelian Cayley graphs, and graph products.