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A girl is moaning about how she is treated for not doing her homework then pauses and says:
やってられないよ
I assume this is a contraction of やっていられないよ which would literally be "I can't be doing it". I've never seen the verb-ている form converted to potential before. Is it a common thing and how does it differ from, for example, やれない in this case?
Maybe I'm barking up the wrong tree entirely. Either way, I don't understand what meaning this sentence is supposed to convey.
As you stated,
「やってられないよ」＝「やっていられないよ」
More informally, you will often hear:
「やってらんないよ」 or even 「やってらんねえよ」 around Kanto (therefore, in fiction as well).
It would probably be better to treat a common phrase like 「やって（い）られない」 as a set phrase rather than breaking it down to understand it.
It simply means "I can't stand it anymore!" You are saying what is happening is ridiculous.
"I've never seen the verb-ている form converted to potential before."
Really? It is used quite often. We say things like:
「つまんねえ[映画]{えいが}！[観]{み}てられないよ～！」"What a boring movie! I can't watch it anymore!"
「[変]{へん}な[曲]{きょく}！[聴]{き}いてられねえ！」"Weird tune! Can't be listening to it!"
「こんなマズいピザ[食]{た}べてらんねえぜ！」"I can't be eating an awful pizza like this!"
Whether you use the present progressive in your translation or not would be up to you.
