Alex, of course you can host your own web site, which is not that hard to achieve.

There are two ways - you can host it of your IP address (which is not a pretty way). Plus, if your IP is dynamic and not static, you're in trouble because your primary web site's address will be changing every time your ISP changes it.

OR you can buy yourself a domain name. It costs less than $10 a year, so its a great investment since you own it, too. I recommend www.GoDaddy.com for all your domain needs. www.Enom.com is not a bad registrar as well.

To setup your own web site, I found this easy to follow step-by-step guide that might help you achieve it:
http://www.dslwebserver.com/

If you are not on Cable or DSL high-speed Internet, your web site's performance will be quite poor.

Hope this helps.