Searching for various examples and counterexamples for sheaves, it is sometimes helpful to look at partially ordered sets with the 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.