Hosting a simple mySQL database online isn't expensive at all... in fact, you can do it for as little as $5 a month. Serverlogistics.com offers hosting and mySQL databases from $10 a month, and GoDaddy.com has simple hosting packages that include a mySQL database for $5 (or less sometimes).
But, if you've got a fixed IP address and an XServe ready-to-go, then you're all set. You need to do the following things:
1) Modify the mySQL user permissions (for whatever mySQL user account you're using to access the databases) to allow connections from external IP addresses. mySQL, by default, only allows connections from localhost (or 127.0.0.1) -- i.e., the local machine.
2) Open/forward port 3306 (unless you've modified the default mySQL port) in your firewall/router.
That's it -- people on the internet can now access your mySQL server from outside of your local network using your fixed IP address. Both of these steps are fairly easy, and step-by-step instructions can be found easily with Google searches.
After those, it's simply a matter of how people access the database. If you'd like a "web browser" to access the database, then you'll probably end up needing to host some php pages on your website (which should ideally be running on the same server as the mySQL server). If via FileMaker Pro, then it would be a matter of your "clients" (the people wanting to access the server) configuring their FileMaker Pro programs to connect to your database.
"Share my database over the internet" is an extremely vague description of what you'd like to do -- there are, literally, thousands of ways to share a database over the internet, and a thousand more ways to access the data in the database. If you could explain, in GREAT detail, what, exactly, you'd like to accomplish, then we can then dig further into this. But, as you've left it, if I may use a metaphor, you've basically brought your car into a repair shop and said, "My car is acting up in a vague and ambiguous way. What, exactly, is wrong with it, and what are the exact steps to take to fix it?" Any mechanic would say, "Buddy, you haven't given us nearly enough information to even get started."