What does the phrase 「こうしちゃおれんな」mean?

こうしちゃおれん is a contraction of こうしてはいられない. It's used by an ojisan-type speaker. ては becomes ちゃ in colloquial speech. おれん is a colloquial negative form of おれる, which is a potential form of おる, which is a little dialectal variant of いる.

こうしてはいられない literally means "I/We cannot be doing this", but it's an idiom that means something along the lines of "It's of no use staying here", "We're wasting time now", "(I just noticed) I must handle this now".