I finally found out what happens when your airplane breaks down while on the ground at the airport. You call the tower and they send someone from Airport Ops to tow you, and you get to sit in the plane during the tow!
I should clarify that the plane didn't actually break down on the ground. It failed its mag check during runup due to a failed left mag, and I couldn't get it to restart after I'd let it die. My club maintenance was immediately able to get it started (although the mag issue was still present) so maybe I just suck at cranking the starter.