Problem with old Ethernet shield

In my research, the MAC is arbitrary and needs to be set in your code:

byte mac[] = { 0xDE, 0xAD, 0xBE, 0xEF, 0xFE, 0xED };

(from Examples -> Ethernet -> WebServer);

After that, Ethernet.begin(mac); should start the interface with DHCP.

I also suggest taking a look at Examples -> Ethernet ->DhcpAddressPrinter as this can help with bringing up the interface.