What hosts a site without you having to pay anything?

You need to be a system administrator to do that. You may configure everything at your house and have you web server running on port 8080. Then you will have to use a service to forward your port back to 80. If your website is not on port 80, people will not be able to access it unless if they know what port it is being hosted on. All Web Browsers connect to port 80 automatically unless it is specified to connect to another port. See Related Link.