Listening for ARP requests is the canonical way to do this. Independent of DHCP or not, any connected computer that wishes to communicate with the outside world will have to make an ARP request for the address of the default router. This request will go out as a broadcast, and contain the source interface's MAC and IP adresses.
If the other computer uses DHCP, it will make an ARP request for it's own address as part of duplicate address detection, which is also a broadcast you can snoop on.
(This works more or less the same way for IPv6, except you need to look for neighbor discovery or router soliciation packets instead.)
Like the answer alluded to, if you have a switch to which you can telnet or use SNMP on, you can extract the MAC table. That will give you a list of MAC adresses on each port in the switch. If you want the IP addresses however, you still need to listen for ARP:s.
On the other hand, if you have access to the default gateway on the network, you can also look at the ARP table there. That will give you MAC and IP addresses for anyone that has recently (for different values of recently...) communicated with it.