Can you become a medical doctor or physician assistant through apprenticing instead of going to medical school?

No. No other route. You were born 100 years too late. No monopoly- pay your dues, go through school and post-grad training like everyone else, and you are welcome. There has to be a way to be sure anyone practicing Medicine really has the knowledge to do so. I had a fellow from one of the med schools in the Bahamas shadow me once- he made a diagnosis of acute prostatitis in a female. Since women do not have prostates, in case you did not know, that was a tad upsetting. You want someone with that much knowledge treating you? I am talking about the US- move to central Africa, there may be no qualifications needed at all.

Where are you going to get your cadaver to dissect? Seriously, that's an important part of the education.

Short answer your question for both is : no (you can not apprentice to be a PA or a MD)

Let's be honest. Medicine isn't something you teach yourself. Do you really think you can teach yourself how the pituitary regulates body functions? What about the Krebs cycle? For either MD/DO or a PA, you have to learn from those who already understand the concepts. And as other have said, you can't sit for the exams unless you've been formally trained?

No, You have to go to med school to be eligible (by state law) to sit for the medical licensing exam. You can do private study all you want, it will NEVER make you a doctor.

You need to have a degree in the medical field before you can be an MD. Through apprenticeship, it might be possible for you to become an assistant of the physician.