I see when I start a new project, it has a company domain. However I don't have my own website. So I have no domain to use. What should I put in this field, if I want to make a legitimate app that Google will accept on their app store? Am I required to have a website prior to posting an app to Google Play store, so that my app will have an actual working domain? Or can I set this to anything I want it to be? Does Google basically require a person to have their own website before they will even consider apps submitted to them?
The domin you see is not a web site. You don't need to have your own website.
The "Company Domain" will be used to generate your package name. Your package name is just a unique identifier for your application in the Google Play Store.
You can use any domain name you want (as long as it is unique) - see more details here.