。
She was dying to know his address.
Literal
She [topic-は] his [genitive-の] address [object-を] wanting-to-know couldn't-stand-it.
~てたまらない ('can't stand X-ing, dying to X, irresistibly X') attaches to the te-form of an adjective (or to ~たい with adjective-style desiderative). 知りたい ('want to know') → 知りたくて ('wanting to know') + たまらない ('unbearable') = 'unbearably wanting to know.' Used for any feeling so strong it overwhelms — hunger, sleepiness, longing, pain. The past たまらなかった captures that this was her state at some past moment.