Instantons are, by definition, solutions to the equations of motion, with finite, non-zero action. For the case of a $\mathrm{U}(1)$ gauge theory, the Euclidean action, over a manifold $X$ is $\newcommand{\d}{\mathrm{d}}$
$$ S[a] = \frac{1}{2g^2}\int_X f\wedge \star f , $$
where $f=f_{\mu\nu}\ \d x^\mu\wedge\d x^\nu$ is the curvature of the gauge field, $a=a_\mu \ \d{x^\mu}$. The equations of motion are simply the vacuum Maxwell equations
$$ \d f = 0 \qquad\text{&}\qquad \d\star f=0,$$
or, defining the codifferential $\d^\dagger := -\star\d\star$,
$$ \d f = 0 \qquad\text{&}\qquad \d^\dagger f=0.$$
But these are exactly the defining equations of a harmonic 2-form. By Hodge's theorem, the space of harmonic $p$-forms on $X$ is precisely the $p$-th cohomology group $\mathrm{H}^p(X)$. Therefore, instantons in a $\mathrm{U}(1)$ gauge theory are counted by $\mathrm{H}^2(X)$.
If $X=\mathbb{R}^4$, which is the case in Srednicki, it is obviously topologically trivial, hence $\mathrm{H}^2(X) = 0$, and thus there are no instantons.
On topologically interesting manifolds, however, you can have $\mathrm{U}(1)$ instantons. For example, on $X = \mathbb{S}^2\times\mathbb{S}^2$, $\mathrm{H}^2(X;\mathbb{R})=\mathbb{R}^2$, so there are instantons labelled by two independent integers$^{(*)}$.
As such, on topologically interesting manifolds you can add a $\theta$ angle term,
$$S_\theta = \frac{i\theta}{8\pi^2}\int_X f\wedge f,$$
as the Pontryagin index is, in this case, non-zero. Note, that this is equivalent to the fact that $f$ cannot be written everywhere as $\d{a}$. The obstruction to write $f$ as $\d{a}$ lies, again, in the non-trivial cohomology. These are all different manifestations of the same thing.
$^{(*)}$the fact that they're labelled by integers rather than reals comes from the fact that flux is quantised and thus it is $\mathrm{H}^2(X;\mathbb{Z})$ counting the instantons rather than $\mathrm{H}^2(X;\mathbb{R})$
