Yes, it's possible to run your own website with any functionality you need, for absolutely no cost, and without any advertising at all.
You simply need to run your own web server.
Here's the steps involved:
1) Create your web site
2) Setup a cheap old computer you have lying around with Apache/ISS and PHP/ASP/J2EE and MySQL/PostgreSQL. Linux or Windows doesn't matter.
3) Get a free subdomain from No-IP or DynDns - they will work with either static or dynamic IPs (with their respective free update clients). If you want to, you can pay for a proper domain name (which go for as cheap as a couple US$ these days).
Voila, you're now hosting your own website with unlimited storage, unlimited bandwidth (if your internet plan is unlimited), unlimited database schemas, full access, all the features you could ever need, and it didn't cost you a cent.
You aren't at the mercy of some shady, unreliable third-party who you KNOW simply cannot survive financially without there being SOME catch; you don't have to be constrained by those stupid "now any idiot can pretend to be a web designer!" WYSIWYG "site-o-matic" editors or "now you too can make a site that looks exactly the same as our other 10 million users" template selectors; and you don't have to build a site on the crumbling foundation of the bane of the internet - social networking websites.
And the best part is, if your site ever becomes popular beyond what you imagined and your private server is no longer suitable (or you no longer have the time to maintain/manage it), you can just find a proper web host that supports the platforms your site is already built on, and just shift your site there without disrupting your visitors.
And the free experience you gain in running a whole web server certainly won't ago astray either. Running your own web server is like all your Christmases at once for a web developer.
EDIT:
Yes, the computer you use as the web server must be left on and running 24/7 and connected to broadband internet. If the computer is off, on Standby, or Hibernate, it won't be serving web pages, that's how computers work. The monitor doesn't need to be on, you don't even need one connected (see: UltraVNC), but the actual PC must remain on.
Do NOT use your regular PC (that you use for internet, IM, email, games etc) - not that you can't, but it's a really bad idea. Your server software will be consuming resources, and the server needs to be kept as clean and functional as possible - the more software on it and the more you use it, the more likely it is to crash or fail, killing your website. Not fun for users visiting your site to find it stop working halfway through, that's how you lose visitors.
More detailed steps as requested:
1) Find a computer to use. It doesn't have to be powerful, anything capable of running Windows XP and a generous amount of RAM will do fine. We're talking the cheapest Core2 Duo or a decent Athlon 64. If you have/can find a spare computer 5 or less years old, it will do fine. It will be running 24/7 so you may want to replace the power supply or add some cooling fans. Again, do NOT use your regular, everyday use PC.
2) Choose your OS. Linux will work wonderfully if you know it, but Windows XP is just fine too. I run Windows XP and assume you have no Linux knowledge, so the remaining steps will focus on Windows XP. Truth be told the steps vary wildly depending on your configuration, so I will also assume you want Apache, MySQL and PHP. That's a popular combination often referred to as WAMP.
3) Download and install the Apache web server (Apache HTTPD). It's free. Browse through the httpd.conf file to configure it, referring to the internet if there's something you don't understand (the file is commented throughout, and the default settings should mostly be fine). Ensure you can reach your site before proceeding (point your browser at http://localhost)
4) Download and install PHP, also free. Search the web for tutorials on how to configure Apache to use it. Make sure you test it before proceeding (create a PHP test file in your site directory, calling phpinfo(), and browse to that page).
5) Download and install MySQL, again it's free. Again refer to web tutorials to get PHP's MySQL connection working, and again test out PHP-MySQL connectivity with a test PHP page. The MySQL GUI Tools also contain some handy tools to administer and query MySQL quickly and easily.
NB: You may be tempted to download a software package called "WAMP", "XAMPP" or similar packages. They come with all the server software in one install, already configured to go. Don't. You lose valuable technical experience if you do. Technical experience that will be immensely helpful in developing and maintaining a web site (even if you eventually pay for hosting) - being able to diagnose problems with your server/3rd-party web host, and work around or fix them, will help out a lot. If you download one, you might as well just setup a MySpace page as the experience you gain and control you have is going to be about the same.
6) At this point, if you don't know HTML, CSS, PHP or SQL, take the time to learn them. W3Schools is a good resource, as is the official PHP manual.
7) Build your website. Do it now, before you bother setting up a domain name or spending any money. Plan what exactly you want for your site - what are your goals? What should your site be about? Will it have user accounts? Paid subscriptions? Spend some time designing it: features, functionality, layout, database schemas, everything. Build it. Test it. Get friends to test it. Get strangers to test it. Do what everyone else does and post the URL (or screenshots or whatever) on Yahoo asking for opinions (you may need to do step 8 first to get it publicly accessible). Make sure you get a web developer to test it for security too, not much fun putting up a site only to have it broken or defaced in the first week.
8) Now you have a web server running and a site ready to go, but unless somebody knows your IP address they can't access it. Even worse if your ISP gives you a dynamic IP address. Also, if you have a firewall and/or a router, your server may not be accessible from the internet anyway.
a)Make sure you configure these appropriately (add firewall exceptions and use port forwarding) until you're able to type your public IP address into a browser and reach your site.
b) Sign up for a free domain name from No-IP or DynDns. You will get a subdomain like yoursite.dyndns.org. Both these sites also offer (paid) full domain names (yoursite.com etc) which are useful if you have a dynamic IP address. If you have a static IP, you can still use those services, but you could also go out and just rent a full domain name from any seller, like GoDaddy.
9) So now you have your website running on your own server, and it's on the internet. What next? Advertise it. Generate the content. Do what you need to do to get visitors. Maintain your site: make frequent backups of all pages and data, setup a password-protected testing section so you can try new things safely. Run a similar setup on your everyday PC so you can use it as a development environment. Your job now is to take care of the site.
Don't fully understand any of the steps I've mentioned? Easy: just google the key terms and you'll come up with plenty of information. You're not the first person to want a setup like this. And if you know how to follow steps and search the internet, it really isn't anything too difficult. If you know how the internet works and can figure out PHP/SQL scripting, setting up a functional web server is a piece of cake.