It would be possible but neither MySQL nor Apache/nginx and PHP are a good fit for a small embedded system like the Humax. That's why the Humax software and the custom firmware components use things like SQLite3, Mongoose and Jim.
You'd probably find it would be very slow and might begin to interfere with the box's day job.
Nobody has compiled those components for the platform either so you would have to do that part yourself.
What exactly are you trying to achieve? Using it as a networked storage device is already possible through the Custom firmware and Samba/NFS packages.
at the moment I have pc running XAMPP with a forum running, basically we just use the forum to store letters we can scan, passwords etc - it's just a lot easier to store and find stuff over multiple pc's