I found a sentence like this: "滅多な事では物怖じしない".
But the meaning of 滅多(careless) make me don't understand this sentence. Then I searched something similar on the website and I found this: "私は滅多に怒りません" which means "I rarely lose my temper".
So does it mean: " He rarely shows cowardice".
Here the context of the first sentence: "面倒見のいい性格で交友関係が広く, 滅多な事では物怖じしない胆力を持つ".