。
She avoided answering my question.
Literal
She [topic-は] my question [to-に] answering-thing [object-を] avoided.
質問に答える ('answer a question') always uses に, not を — questions in Japanese are 'answered to,' not 'answered.' To make the verb phrase the object of 避ける ('avoid'), it's nominalized with こと. The full structure 答えることを避けた layers neatly: [answering-the-question] becomes a noun phrase, which then takes を to mark it as what was avoided. A more conversational alternative would be 質問に答えなかった ('didn't answer'), but 避ける specifies that the avoidance was deliberate.