Could it be a problem with security protocols? Modern routers (normally) use WPA/WPA2.
Requirements for WPA can be found in the article AirPort Card: When You Can Join a WPA Network - Apple Support. However, there appear to be somewhat confusing and conflicting reports on the Internet about whether WPA actually works with the older 802.11b AirPort card (that is, not the 802.11g AirPort Extreme card). The 2002 eMac model could only be equipped with an AirPort card (later models used AirPort Extreme).
If possible, one test would be to (temporarily) set a router to no encryption (or perhaps WEP). If the computer then connects, it indicates that the problem is related to the protocol/password.
Sometimes, a good alternative to a built-in card can be a wireless Ethernet bridge (also known as a gaming or media adapter) connected to the Ethernet port of the computer. There are such devices from several manufacturers (for example, WET610N from Linksys and WNCE2001 from Netgear). A wireless Ethernet bridge does not need special drivers, and can be used under almost any operating system. A modern device would support the latest security standards. Configuration is typically done via a web interface (web browser).