Trying out a couple of distributions (in June 2010) I found:
Tinycore (2.11): Booted from a pen drive and ran ok in text mode. If allowed to run X it ended up with a blank (dark) screen with an arrow cursor in the centre.
Puppy (5.0.1): Booted up from an attached USB CDROM drive. The desktop didn't appear in all its glory - but the menu bar was there at the bottom along with the Menu button. (I assume this restricted view is due to the limited amount of RAM. 20M Free is shown on the menu bar).
I could run things via the start menu - such as the Internet Setup Wizard, but everything happened very slowly thanks to the USB1.1 interface. Things should be much quicker if you boot from the IDE interface!
In June 2014 I had an email from Jochen who had used the t5300 with an 8GB Compact Flash card. He was running router and server software (without X) from:
eisfair.org Eisfair is a quick and easy-to-install web-server that uses Linux as its underlying operating system. It's installation as well as its operation requires no basic knowledge of Linux. fli4l.de FLI4L is a Linux-based ISDN, DSL and Ethernet router that can be installed in the basic equipment on a single floppy disk. As a platform a 486 with 16MB of RAM is more than adequate for this.