Obviously what is meant is that they have work available that can be had on a part-time basis if you wish. If all you can afford to give is weekends or a few hours each day (after your regular job, for example) or wish to become a manager who actually lives in an apartment on the premises and runs that location's office, that may be available as well. U-haul also allows franchises (so I have observed) that allows you to set up a rental at your own business, so whatever the hours of operation are for your business, then that is what is implied. If at your own place of business you run three shifts, then you could indeed offer the U-Haul rentals any hour of the day.
Don't ever make the mistake of assuming a job offer is going to be exactly what you may be used to doing, as far as hours. I have seen jobs that are all hours of the day and night and almost any sort of rotating days, etc.
Lastly, don't be ridiculous. No human has a job that is 24/7/365. Even machines are stopped occasionally for servicing.