Hi Nigel,
eMac (USB 2.0) - Technical Specifications
https://everymac.com/systems/apple/emac/specs/emac_1.25.html
I guess that it would be correct to assume that you have tried to set up a wireless connection at this point. Had your eMac been equipped with the (optional) internal 802.11g AirPort Extreme card, or are you using another type of device (such as a USB Wi-Fi adapter attached to the eMac)? In order to successfully connect to a Wi-Fi router, the security protocol (WPA, WPA2, etc) must be OK for all involved machines. Any information about this? Which type/model router is used?
Also, if the machines are near each other, you may want to use an Ethernet connection instead (via the LAN ports of the router).
You mentioned OS 9 on the hard disk. This would be via the Classic environment for this very eMac model (no booting into Mac OS 9).
Furthermore, it would be interesting to know the type/model/year of the other computer (the iMac) and its operating system version. This in order to verify networking and file sharing compatibility, and possibly to suggest alternative solutions.
https://everymac.com/systems/apple/imac/index-imac.html