Sie hängen mir alle zum Hals raus.

English Translation

I am sick of them all.

Is this idiomatic? I can’t quite make sense of a literal translation.

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Yes, it’s idiomatic.
The Idea is, that you got more of something (maybe originally food, but later i could be anything) than you can swallow.
The nearest English expression that comes to my mind is “I am fed up with something”.