How does the AirPort connect to the internet? If you use DSL, Cable Modem, or PPPoE, make sure the ethernet cable from the modem to the AirPort is connected to the WAN port (marked with a dotted circle). The G5 should be connected to the AirPort via Ethernet through the LAN port (marked with the double arrows).
Also, open up your base station using AirPort Admin Utility (in /Applications/Utilities). Then:
1.) Click the "Show All Settings" button on the left.
2.) Click the "Networking" tab.
3.) Make sure "Distribute IP Addresses" is checked.
4.) Select "Share a single IP address (using DHCP and NAT)"
5.) Click the Update button. Wait for the AirPort to restart.