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.
You can get insurance through the national marketplace, this was put in to effect with the ACA. The biggest problem with that though is the cost can be insane (single person, $800+ month). Usually why people go through the company is it is much cheaper (I pay about $60/month premium for just me).
It is usually cheaper because either a company "self funds" or they have a large number and better leverage with insurance companies. If you're a business owner, you work with brokers who will try to find you the plan that best suits you or your company's needs. Finally retirees will either have insurance through their pension, or will use Medicare which is another government provided insurance for people over 65.
I know that probably raises more questions but hopefully that's a start.
Where I am, companies provide protection which are generally pretty okay. But everyone still gets personal ones, and they do not cost too much. Perhaps, 50-100? And tend to cover the gaps the organisation's sometimes don't.
Oh, and you do not have to pay extra for the company's. It is part of your remuneration package. (we get paid less though).
To add to this, always contact the hospital and ask for an itemized bill. Getting that bill can be the difference between owing hundreds of dollars vs thousands.
Some hospitals will also have different scales for how much you have to pay based on income. Make sure you find out.