Why do some people think that they can start an online business without having a website?
Because you can.
They may have an online store hosted on someone else's platform. They may work by word-of-mouth so their network is all they need. They may use a social profile; with their skills, they can turn that profile into a lead-generating machine.
Many new business owners think the internet is a save-all for business. What's going to happen when you don't have it? It's a tool with an expiration date just all of the previous technologies we've used in the past. Plenty of people are making money without a Google Profile. A website is okay, but it's not the end of the world.