What sort of emergency do you envision? Most of today's financial emergencies hinge on either losing your job or incurring extreme debt due to uninsured medical care. I can't speak for all BI proponents but I, personally, support universal health care in addition to BI.
With health care taking care of medical expense emergencies, basic income is left to take care of job-related emergencies. Your example of "my car broke down" would seem to fit that category. So what then? Joe loses his job but basic income will make sure he never goes homeless and always has enough food to eat. Instead of having a major crisis he can take a break from working and maybe upgrade his skills.