Yes, individuals and self-employed persons can buy health insurance on the health insurance marketplace. Since you're one person buying a plan you don't have the ability to negotiate with health insurance companies the same way a corporate employer buying policies for many people can negotiate. This means individual plans are expensive and coverage is often lacking, but it's still better than ending up with cancer or a massive car wreck and being without insurance.
There are also government funded plans like Medicare, Medicaid, etc and I have never dealt with those. Have been a self-employed contractor though. A policy I paid $600/mo for as an individual, self-employed contractor costs about $200/mo for a plan with better coverage for both my husband and I through his corporate employer plan.