For those who are interested:

Oracle Support has informed me that the Oracle Net8 Client version 8.0.4 is not supported under Windows 98 (ie. it doesn't work). 8.0.4 is the current latest release of the Oracle Net8 Client.

Oracle Net8 Client 8.0.5 will work with Windows 98 but is yet to be released (I'm not sure when it will be released).