What does it mean for a problem to be both NP hard and coNP hard
A language $L$ is NP-hard if for every language $R$ in NP there exists a function $f$ computable in polynomial time such that for all $x$, $x \in R$ iff $f(x) \in L$.

A language $L$ is coNP-hard if for every language $R$ in coNP there exists a function $f$ computable in polynomial time such that for all $x$, $x \in R$ iff $f(x) \in L$.

If a language is both NP-hard and coNP-hard then its exact complexity lies above both NP and coNP. Indeed, it is conjectured that NP$\neq$coNP, and this implies that a problem which is both NP-hard and coNP-hard belongs to neither NP nor coNP.

Problems which are both NP-hard and coNP-hard do exist. For example, any PSPACE-complete problem is both NP-hard and coNP-hard.