Short answer "No"
Longer answer, you probably could but it would be pretty challenging and more of an exercise to prove it could be done than a practical implementation of MySQL.
The stock GFH runs the Axentra Hipserv OS loaded into flash memory. That OS is based on Red Hat Linux and runs a web server and the minidlna database / media service.
Here's a link showing how to install ArchLinux on the GFH http://blog.philippklaus.de/2011/04/install-archlinuxarm-on-the-seagate-goflex-home/ It also gives the GFH Hardware architecture. ArchLinux is perhaps a better option than Hipserv for MySQL.
This link http://dev.mysql.com/doc/refman/4.1/en/supported-os.html lists the OS's that MySQL is known to run on.
So if you really know your way around Linux and are up for a challenge, you might be successful in getting MySQL running. But it would be a pretty restricted installation. You wouldn't get the Workbench so you'd have to manage it through command lines in a terminal.
If you want a standard MySQL server implementation, with the management Workbench you really need a full Linux, Windows or Mac OS. Using an old Laptop running Linux as a server would be a much easier option (I've got a very old IBM Thinkpad 300 running Ubuntu though lighter distros work much better).
cheers
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For the sake of your own health and sanity, if your data is important to you, keep a second copy on another storage (a backup)
