Ok, there are two different settings that involve DHCP, one being the public IP address that is assigned to the AirPort and the other being the IP addresses that are assigned to the computers on the private network.
Before doing the following, I would hard reset the AirPort Extreme by holding the reset button on the back for 5-10 seconds, just so we can get a default onfiguration.
If you run AirPort Utility, go to Manual Setup, and click on Internet your Connection Sharing at the bottom should be listed as Share a Public IP Address. This enables your AirPort to be the gateway and to receive a public IP address from your cable provider. This should really already be enabled since you did a hard reset.
Now test to make sure you have an internet connection. If not, then you need to reset your cable modem. If after resetting your cable modem you still can't get internet connection then you need to call your cable provider and have then do whatever they need to do to get you a new IP address.
Now that you have a working internet connection you need to setup your private network. Run AirPort Utility, click on Manual Setup, and go to the Internet tab again. Write down the DNS servers you see listed. For Configure IPv4, you need to change the setting from Using DHCP to Manually. Enter the IP address as 192.168.249.10, the subnet mask as 255.255.255.0, the router address as 192.168.249.10, and then use the DNS Servers you just wrote down.
Now click on the DHCP tab next to the Internet Connection tab and select 192.168.249.100 as your beginning DHCP address. Ending address could be something like 192.168.249.200. Update your AirPort and then see if things are working correctly.