。
She was aware of being stared at fixedly by a stranger.
Literal
She [topic-は] unknown person [agent-に] fixedly was-being-stared-at [nominalized-こと] [object-を] was-aware.
Several layers stack here. 見知らぬ uses the classical/literary negative ぬ on 見知る ('to know by sight') — modern Japanese says 見知らない人 in conversation but 見知らぬ人 has crystallized as the default written form for 'a stranger.' じっと is a state-mimetic for fixed, unwavering attention; pair it with 見つめる ('to gaze') and you get the prickly, prolonged stare that motivates the whole sentence. Then comes the syntactic spine: 見つめられている is passive + ongoing ('is being stared at'); ~ことを意識する nominalizes the whole event with こと and feeds it into 意識する ('be conscious of') as a direct object. The agent of the passive is marked with に, the standard pattern for 'stared at by [someone].'