Searching for various examples and counterexamples for sheaves, it is sometimes helpful to look at partially ordered sets with poset topology: a set $U$ is open if and only if $x \in U$ and $x < y$ implies that $y \in U$.
But for such topological spaces the category of sheaves of abelian groups has enough projectives. The sheaf $Z_{\ge x}$, the extension by zero of the constant sheaf on the open set $\{\ge x\}$, is projective. Indeed, $\operatorname{Hom}(Z_{\ge x}, F) = F(\ge x) = F_x$, the stalk at $x$, so $\operatorname{Hom}(Z_{\ge x}, -)$ is an exact functor.
And any sheaf $F$ is a quotient of direct sums of these: take one sheaf for each element of each stalk $F_x$.
This is essentially the same as David Treumann's answer. The sets with the unique smallest open neighborhood property, also called Alexandrov spaces, are the same as posets once you identify the topologically equivalent points.
