An individual can sign up for a plan through their State’s health insurance exchange or the federal government’s HealthCare.gov website.
It is usually more expensive than getting it through an employer - but works to serve small business owners, freelancers, etc.
A few States (like Massachusetts) have semi-universal systems that cover all individuals that earn under 150% of poverty, independent students, newly unemployed, etc.
A lot of Americans are also covered under Medicare, Medicaid , Social Security and other programs.
Retirees aged 65 and older are eligible for Medicare - a semi-universal federal system that covers pretty much everything and accepted most places.