Medicare and Medicaid would still exist if Obamacare was repealed.
Private health insurance still does exist (e.g. Cigna). And more insurers would pop up to fill market demand.
Cash on the barrel isn’t a bad idea for the long term either. Producers would pay and there are plenty of doctors who would lower their rates if paid in cash rather than having to deal with insurers.
A good example of a private market that works is the deal cut by Pennsylvania Amish with local hospitals. Their cash rates are about 1/3 what everyone else pays (a key reason being they don’t milk the system and they don’t sue).
For those Amish who can’t afford the local doctors, there’s a physician down in Mexico who set up a practice specifically tailored for the Amish. Once again, all cash. No insurance.
Also, Medicare and Medicaid are massive government distortions on the market, so don't pretend that it is some sort of free or private market when the two largest entities involved are run by the government. One of teh main reasons healthcare was so expensive was private insurers and cash payers subsidize the low medicare/medicaid fees by paying higher fees.