The difficulty in going car free usually revolves around the 5% of the time that is impacted by an unusual or occasional circumstance. Personal injuries (yours), medical emergencies (kids at school/daycare), tight schedules (anybody's), visiting relatives (arrived by train/plane), unplanned job travel/commute to a distant office, large purchases, etc. Sometimes these can be managed via a rental car, a taxi, public transportation, a family member, or a friend.
All in all, my goal is not to try to control the uncontrollable. Instead, I try to plan financially. I understand that over the course of a year, I won't always be able to keep non-bike transportation expenses to $0. Instead, I track those expenses closely. As long as my transportation expenses significantly beat my personal car-cost average, I'm satisfied (yes, I do want to beat the car-driving numbers by quite a bit because bike commuting sometimes carries a convenience cost that servers as a strong motivator in addition to the fun and health benefits).
So most likely a short-term issue (less than two weeks) would be public transportation or a rental car. For a 3+ months without a public transportation option--yeah, I would probably have to pick up a cheap ride off CL and resell it. Without an available carpool or public transportation, it doesn't take long to rack up the costs for car rentals and taxis. But then, I haven't had such an issue in my entire life, and I've only missed a few unexpected days of work in the last 10 years. Conflicting schedules impact me more often.