We present a sequent calculus for the modal logic $\mathbf{S4}$, and building on some relevant features of this system (the absence of contraction rules and the confinement of weakenings into axioms and modal rules) we show how $\mathbf{S4}$ can easily be translated into full propositional linear logic, extending the Grishin-Ono translation of classical logic into linear logic. The translation introduces linear modalities (exponentials) only in correspondence with $\mathbf{S4}$ modalities. We discuss the complexity of the decision problem for several classes of linear formulas naturally arising from the proposed translations.