Is it feasible to become a paramedic full time and do a PhD in an unrelated field?
A does sound impossibly difficult to me.  The first few years of a masters/PhD sequence are usually based on coursework.  You will need to attend classes regularly on weekdays, which seems like it would probably be incompatible with a paramedic's work schedule.  Even beyond coursework, particularly in lab sciences, you may find that parts of your research have to be done on a particular schedule.

Moreover, at all stages of graduate study, you should expect that your coursework and/or research will require a time commitment at least comparable to a full time job (40 hours per week), likely substantially more.  Many graduate students struggle with the workload, even without added commitments.  Adding a full time job, especially one with as much stress, long hours, and sleep disruption as a paramedic must have, seems to me like a burden well beyond what any human could be expected to successfully bear.