if the double arrows show that X and the error term are correlated, but that neither variable affects Y, is endogeneity a problem in this scenario? Why or why not?
Endogeneity is a problem when you have a regression model
$$E(\epsilon | X) \neq 0$$
But if you assume that neither $X$ nor $\epsilon$ affects $y$ then the above model is mis-specified. In other words, $y$ is not a function of $X$ or $\epsilon$ but rather of something else. So there is no point in discussing $X$ or $\epsilon$ when talking about $y$.
Endogeneity can only be discussed conditional on the above relationship ($y=X\beta+\epsilon$) being true. Otherwise (for example, if $y=W\gamma+\nu$), then you need to discuss endogeneity in terms of $W$ and $\nu$.
When you say a variable is endogenous, you need to specify "endogeneous with respect to what other variable?". I in case of $y=W\gamma+\nu$, if $E(\nu|W)\neq0$, i.e. $W$ is endogenous, then $W$ is endogenous with respect to $y$.