1) In a similar spirit to your own example, here is a necessary condition for a sheaf of abelian groups $\mathscr{F}$ to be an $\mathcal{O}_X$-module for *some* scheme structure on $X$:

For each point $x \in X$, the stalk $\mathscr{F}_x$ has to admit a module structure over a local ring.

This is not too explicit, so one can replace "has to admit a module structure over a local ring" by some more concrete condition: the one that comes to mind is

Torsion part of $\mathscr{F}_x$ has to be a $p$-power torsion for a prime $p$ (in fact, $\mathscr{F}_x$ has to be uniquely $\ell$-divisible for every prime $\ell \neq p$)

Although this has no ambition of being close to sufficient, it already excludes a lot of sheaves (locally constant sheaves with a not-$p$-power torsion, for example).

2) The following necessary condition is somewhat similar, but this time it is an honest criterion for being quasi-coherent. The drawback is that it seems hard to check in general. One can use the fact is that for a quasi-coherent sheaf $\widetilde{M}$ on $X=\mathrm{Spec}\,R$, the canonical maps $\widetilde{M}(X) \rightarrow \widetilde{M}_x$ and $\widetilde{M}(X)\rightarrow \widetilde{M}(D_f)$ are localization morphisms. This property can be somewhat-accurately described without referencing the ring $R$, i.e. purely in language of sheaves of abelian groups, in the following way.

**Lemma.** Given an $R$-module $M$, where $R$ is a commutative ring, for $r \in R$ denote $r_M=[r\cdot -]|_M,$ i.e. the endomorphism of the abelian group $M$ given by multiplication by $r$ on $M$. For $S \subseteq R$ multiplicative set, denote $S_M=\{r_M\;|\; r \in S\}$. Then $M \rightarrow S^{-1}M(=M\otimes_RS^{-1}R)$ and $M \rightarrow S_M^{-1}M=M\otimes_{\mathbb{Z}[S_M]}S_M^{-1}\mathbb{Z}[S_M]$ are canonically isomorphic (as objects in the coslice category $M/\mathsf{Ab}$).

Call a map of abelian groups $M \rightarrow N$ *endo-localization* if $N$ is of the form $N=S_M^{-1}M$ for some subset $S_M \subseteq \mathrm{End}(M)$ of pairwise commuting endomorphisms (and the map $M \rightarrow N$ agrees with the localization map). Then

If $\mathscr{F}$ is a sheaf of Abelian groups on $X$ that is a quasi-coherent sheaf with respect to some affine scheme structure on $X$, then

Each of the maps to stalks $\mathscr{F}(X) \rightarrow \mathscr{F}_x$ is an endo-localization, with respect to multiplicative set of endomorphism containing (multiplication by) all but possibly one prime.

There is a basis of open sets $\{U_i\}_i$ of $X$containing $X$ such that $\mathscr{F}(U_i) \rightarrow \mathscr{F}(U_j)$ is an endo-localization whenever $U_j \subseteq U_i.$ with respect to multiplicative set generated by a single endomorphism.

**Proof of Lemma:** Both sides are/can be interpreted as:

1)$R$-modules on which $S$ acts by isomorphisms (and $R$-module map to it): For $ S_M^{-1}M$, the module structure is given on the left-hand side, i.e. $r(m\otimes f)=(rm)\otimes f$, and this makes the map $M \rightarrow S_M^{-1}M$ an $R$-module homomorphism. Also for $r \in S,$ $r(m\otimes f)=r_M(m)\otimes f=m \otimes r_Mf,$
showing that $r$ acts by isomorphism on $S_M^{-1}M$ (with the unique preimage to $\sum_i(m_i \otimes f_i)$ given by $\sum_i(m_i \otimes r_M^{-1}f_i)$).

2) $\mathbb{Z}[S_M]$-modules on which $S_M$ acts by isomorphisms: For $ S^{-1}M=M\otimes S^{-1}R$, let $S_M$ act on the left factor, and use $r_M(m\otimes f)=(r_M(m))\otimes f)=rm\otimes f=m\otimes rf$

Now use the universal properties of $S^{-1}M, S_M^{-1}M$ with respect to 1) and 2), resp., to conclude that these are canonically isomorphic.

anyfield. This just doesn't feel like the right question to be asking. $\endgroup$ – Tim Campion Apr 14 '19 at 17:23