I was at the point of walking out when he suddenly spoke to me.

"be at/on the point of doing sth" 是一个固定短语,意思是"即将做某事、正要做某事",用来描述某种动作或事件马上要发生、但尚未发生的那一刻。 其中 at the point of 是核心结构,of 后面接动词的-ing 形式(doing sth),强调动作迫在眉睫的状态;前面可以用 was / were 等系动词,搭配主语的状态,也可与 on 互换表达同样的意思。