。
She felt something that was neither love nor hate.
Literal
[topic-は] love [neither-とも] hatred [nor-とも] cannot-determine thing [object-を] felt.
~とも~ともつかない is a literary expression meaning 'cannot be determined as either X or Y' — a way of describing something that resists categorization into two opposites. Each repetition of とも attaches to one alternative; つかない is the negative of つく used in the sense of 'to be settled/determined.' The pattern shows up especially in narrative descriptions of ambivalent emotional states: 笑いとも泣きともつかない ('something between laughter and tears'), 夢とも現実ともつかない ('neither dream nor reality'). もの ('thing') is doing nominalizing work here, packaging the unnamed feeling into a grammatical object.