How do I get over the fear of making a costly mistake at a new job?

Being new and scared around expensive parts and dangerous machines is not a flaw. It is your mind taking the work seriously. The careless ones are the ones who get hurt. Your fear, kept in check, is actually keeping you safe right now. The fear will shrink the only way it ever does, through repetition. Every part you run right, every shift you finish with all ten fingers, teaches your body that you can do this. You are not supposed to feel calm yet. Calm is earned over months, not handed to you on day one. Give yourself that time. As for the older guys, remember they were once as green as you, even if they will not admit it. Do not wait for them to be warm. Watch what they do, ask short clear questions, and let your steady work earn their respect. Focus on the machine in front of you. Slow, careful, correct. The confidence comes after the competence, not before.