I don't think that the example you chose is the simplest one. It might be better to start with say a vector bundle $\mathcal V \to X$ over some base space $X$. Suppose now that $G$ acts on $X$. A $G$-equivariant structure on $\mathcal V$ is a choice of $G$-action on $\mathcal V$ (assuming it exists) preserving the vector bundle structure and compatible with the given $G$-action on $X$.
E.g. if $G$ acts on a space $X$, and $\mathcal V$ is functorially assigned to $X$, e.g. the tangent bundle or cotangent bundle, then $\mathcal V$ will have a natural $G$-equivariant structure.
Another example: if $X$ is $n$-dimensional projective space, with its usual action of $GL_{n+1}$ (which factors through $PGL_{n+1}$) and $\mathcal V$ is the tautological line bundle, then $X$ is naturally $GL_{n+1}$-equivariant. To see this, note that (if we remove the zero section) then the tautological line bundle over $\mathbb P^n$ is just ${\mathbb C}^{n+1}\setminus \{0\} \to ({\mathbb C}^{n+1}\setminus \{0\} )/ {\mathbb C}^{\times}\cong {\mathbb P}^n$, and $GL_{n+1}$-acts compatibly on the members of this diagram.
Another way to phrase the same structure on $\mathcal V$ is to say that for each $g\in G$, there is a given isomorphism $\alpha_g: \mathcal V \cong g^\*\mathcal V$ such that for $g,h \in G$, one has $h^*(\alpha_g) \circ \alpha_h = \alpha_{g h}$.
This latter condition can then be translated to a corresponding condition on the sheaf of sections of $\mathcal V$, which finally can be translated to a condition on any sheaf.
What this amounts to is that one has an action of $G$ on the sections of the sheaf (let's call it $\mathcal F$) compatible with the $G$-action on $X$. The only thing one has to be careful about is that since the sheaf may not be determined by its global sections, one has to think about sections over arbitrary open subsets $U$, and then one has to take into account the fact that $U$ may not be $G$-invariant.
So putting it all together, one gets for any $U$ open in $S$ an isomorphism $\alpha_{g,U}: \Gamma(U,\mathcal F) \cong \Gamma(gU,\mathcal F)$ (which is the action of the element $g \in G$), compatible with restriction to open subsets, and such that $\alpha_{g,h U} \circ \alpha_{h,U} = \alpha_{g h, U}.$
Caveat: I hope I have my composition formulas correct, but if I've gotten things tangled up, I'm sure they'll be corrected.
As for your original question, you should specify in what topology you want to consider your sheaves (Zariski or etale are the two possibilites that come to mind), and perhaps also what kind of sheaves you are thinking about (e.g. locally constant etale, coherent, ... ).
E.g. the structure sheaf of Spec $L$ has a natural equivariant $Gal(L/K)$-structure, just given by the $Gal(L/K)$-action on $L$. (But note that the terminology here is much more complicated than the actual facts being discused: Spec $L$ is just a point, and so the $Gal(L/K)$-action is necessarily trivial on this point, and giving a sheaf is just the same as giving an abelian group (or set, or ..., depending on what sort of sheaves we are discussing). So giving a $Gal(L/K)$-equivariant sheaf is just giving an abelian group (or set, or ... ) with a $Gal(L/K)$-action.)