Section 1501 of the Patient Protection and Affordable Care Act (PPACA or just ACA) of 2010 also known as Obamacare requires you to "maintain minimum essential coverage." If you do not, you will have to pay a penalty. For 2016, that penalty is at a minimum $695 per adult. The idea is that if you do not voluntarily participate in the "shared responsibility" to keep costs down, you will be forced to contribute in this other way.
However, there are a number of ways you can avoid the penalty. You can get an exemption. Some exemptions have to do with income levels. Others have to do with hardship and several other situations. The interesting one is membership in certain groups. Members of recognized tribes are exempt. Certain religious sects, e.g. the Amish, object to insurance of any kind including Social Security and Medicare. You can also get an exemption if you participate in a recognized "health care sharing ministry."
Health care sharing ministries are not-for-profit member organizations created to share medical costs among their members. There are only about a handful of them and there will be no new ones unless the law is changed. To be recognized, they have to have been in continual operation since 1999.
Because they are not insurance companies, they are not regulated by insurance laws. This is what allows them to discriminate when accepting participants. By accepting only healthy participants without pre-existing conditions, avoiding the ones who need it the most, and excluding common procedures, their costs are much lower than insurance companies who are required to accept everyone and adhere to minimal care standards.
Thus, if you can get past the moral dilemma of this supposedly Biblical practice, you can save substantial cash without a penalty and still have basic health care coverage. Each of them have their own rules usually spelled out in a document called guidelines or something to that effect. You will need to examine the document carefully to determine if it meets your needs. It is usually quite simple and in relatively plain language.
Some of the major players in this arena are:
Some new ones have emerged on the scene lately, but it is unclear how they get around the continual operation provision of the PPACA.
In this age of Uber, ZipCar and AirBNB I am sure people are tinkering with ways to make health care sharing the norm rather than the exception to avoid contributing to the profits of insurance companies and their shareholders. I would personally like to see a cost sharing organization that would accept everyone and cover all common procedures as they have need. That could be accomplished by transforming an existing organization or change the law to allow new ones. In the meantime, if you were to assume the burdens of everyone as well as the ones who shirk their responsibility, that would truly be Biblical.