Self insurance for an individual would require stashing away enough cash to cover a serious medical problem. The heart attack my neighbor had last year that resulted in $200,000 in medical bills, but that is far from the worst case. The ACA could have created special provisions for the wealthy to self insure, but didn't. It would have applied to very small percentage of the population and require some form of vetting to assure they actually had sufficient assets to self insure. I suspect that almost everyone in the top 1% is insured via work or some corporate entity they set up for doing business.
If Bill Gates wants to pay cash for medical care, he is welcome to do so. He can buy a Bronze plan and never use it to avoid the penalty.
There are, of course, other options. Most people do not realize that most medical expenses are negotiable, they really aren't set in concrete prices and fees. I don't know how that could be addressed through legislation such as the ACA bill, but it is true.
Anecdotally, I have had several relatives who worked as case managers, and various other ancillary medical fields as well as within medical centers as physicians and nurses who didn't carry (sufficient) personal health insurance. (Most of these cases are a few years/decades in the past) who had major medical issues arise (heart surgeries, emergency gall bladder surgeries, extended hospitalizations, etc.) that initially resulted in very large billings. They contacted the hospitals, direct care providers and other assorted providers and negotiated their bills to a tiny fraction of the original charges. Many of the bills were written off entirely. I'm not saying that this is possible in all cases merely that because these people are intimately aware of the workings and processes of how billing for care functions within these systems they were able to negotiate bills that were initially in some cases hundreds of thousands of dollars, down to pennies (or at least dimes) on the dollar. This type of situation is not available to everyone, but the point of me mentioning it, is that perhaps there should be a larger variety of coverage offered/acceptable to meeting ACA requirements.
Perhaps allowing an individual to create a personal health savings account with a secured deposit sufficient to annually cover relatively less expensive emergencies and routine/preventative care issues, and a catastrophic policy that does nothing but serve as back-up for those unexpected but relatively rare major events. This isn't going to be an option for the lower end wage earner, but a much larger portion of the more affluent mid-upper middle class professionals might find such an option attractive, at least while they are younger. Of course, such choice options, may require additional public funding to make up for the withdrawal of offsetting revenues from the system, and that tradeoff (more choice means more tax dollars) should be made a clear consequence of the options for more self-insurance options within the system, perhaps even directly targeting those who choose such self-insurance options.
Most of the complaints seem to revolve around the narrowed options currently allowed, and as the legislation matures, it would seem to be reasonable to create a framework where it could evolve into offering a more diverse set of coverage that allows for more options and personal choice.