masto.host is not affiliated with the Mastodon project. We link to their site from joinmastodon.org without receiving any compensation or profit from it, because I believe that being able to kickstart a new Mastodon server is good for the network.
If you have a domain name and an always-on machine connected to the internet from your home, and an internet service provider that allows hosting websites from your home internet connection, then you can technically host Mastodon for free. However, that is not a common scenario and has many disadvantages. You can’t get around needing a domain name and a powerful enough machine with a fast internet connection to get good user experience. That costs money. Which is why no free Mastodon hosting is possible.
However, most servers allow you to sign up on theirs for free.