。
She refused to take up the position.
Literal
She [topic-は] that position [object-を] take-on [won't-try-to-ようとしなかった].
引き受ける is a compound verb literally 'pull-receive' — meaning 'to take on, accept (a responsibility, task, or position).' ~ようとしない ('won't try to / refuses to') is stronger than the plain negative — it implies stubborn unwillingness rather than mere absence of action. 地位 ('position, status') typically refers to a formal role or social standing, distinct from the more general 立場 ('standpoint, situation'). Past form しなかった retroactively casts the refusal as a fact: she didn't, and wasn't going to, take it on.