To run a UNIX based web server, you will need a type of UNIX, and Sun's Solaris will work wonderfully for the operating system. You will also need a permanant Internet connection like DSL or a cable modem. You will also need a web server application, which is where Apache comes into the picture. You can either download and compile it, or I beleive it is installed by default with Solaris 8 (or RedHat 7.x). Once you have your web server built and you can pull up a page in a browser by entering http://localhost in the address bar, you then need a DNS entry to point requests to that domain name to your local IP. You can install and run your own DNS servers, but there must be 2. Another option is to use someone elses DNS, I have used http://www.zoneedit.com with success.
Good luck. You will learn a lot, but it will be worth it when finished.