It takes a while to get replies on here sometimes.

Yeah, I don't know. No offense, but he sounds shady.

I know (work with) men who are "on-call," but they would all have a car because the nature of jobs where you have call days are such that you need to be able to have reliable transportation, or you cannot do the job. I realize not every contract is the same as where I work (or other facilities like mine), but it would surprise me if an employer would hire someone for an on-call gig when they cannot provide proof of reliable transport (taxis don't count). I am of the opinion that he's not being truthful with you. With the issues you've mentioned, I think you deserve better. Not just because we all deserve better, and not just for your own well-being, but also for your peace of mind.