Presentable module is defined in nLab (https://ncatlab.org/nlab/show/presentable+module) as “the cokernel of a homomorphism of free modules”.
What I’m confused about is that, according to https://en.wikipedia.org/wiki/Free_presentation, given a module $M$ over $R$, “a free presentation always exists”, so shouldn’t that mean any module is presentable?
So what would it look like for a module not to be presentable? Are there particular constraints on what sequences “count” that I’m missing? Or maybe $M$ isn’t always a “cokernel” in these sequences? Or maybe I’m muddling two different definitions?
Put another way, in the definition of a quasi-coherent sheaf (https://en.wikipedia.org/wiki/Coherent_sheaf), we require the existence of a certain exact sequence $$ \mathcal{O}_X^{\oplus I}|_{U} \longrightarrow \mathcal{O}_X^{\oplus J}|_{U} \longrightarrow \mathcal{F}|_{U} \longrightarrow 0 \,, $$ but by the above, given that $\mathcal{F}|_{U}$ is by definition a module over $\mathcal{O}_X|_{U}$, shouldn’t such a sequence always exist?
The issue here, as @kevinarlin pointed out, is I was confusing two notions of "modules": a module over a ring (an abelian group with a ring of scalars) and a sheaf of modules (aka a module over a sheaf of rings, which is a certain sheaf whose sections are modules of the first type).
These notions coincide when $X$ is a single point (since all the information there is is a single module, so we can "forget" the point and just look at the module over it.)
The nLab article was using the second definition (which @qiaochu-yuan points out is somewhat confusing).
The statement "there always exists a free presentation of a module" is only true of the first definition, not of the second, which corresponds with the existence of non-quasi-coherent modules.
So the exact sequence I wrote at the end was of the "second" type of modules, which is why it's not vacuously true.