I absolutely agree that you shouldn't tell them to "calm down" unless you want to guarantee that they won't.
I like the idea of repeating to them their problem as you understand it. That immediately tells them that you have listened to them. Then, I think you can offer them a seat or excuse yourself while you take whatever action you can. Let them know what you're doing.
"That sounds very frustrating. Let me make sure I'm clear on the situation. As I understand it, you're disputing a late penalty because of extenuating circumstances. Dr. X took off the regular late penalty because you submitted your report two days late. However, the assignment was late because you lost power and internet due to the snowstorm and were unable to reach him or get to campus. You've already talked with him and he says his decision is final. You'd like to appeal your grade on that assignment. Is that right? Okay, you'll need to talk to the departmental chair. I think he is in a meeting right now, but I'll call his assistant, Shirley Ulissen to find out. If he's still in the meetings, she'll be able to schedule an appointment for you. If you'd like to have a seat, I'll give her a call right now." Then look away from the student (and down at your phone, directory, whatever) and start doing something so she knows you are taking action.
I do think that most people who are upset in a case like this mostly want to be heard and understood, have someone acknowledge their anger/sadness, and then have the assurance that someone is taking action on their behalf.