I have an Arduino Ethernet Shield and would like to put it to use. However, there is no MAC address printed on the circuit board.
Is there a way to obtain this MAC address or am I to assume that all network access to this board is via DHCP?
You just need to make one up. Any address is fine as long as it's unique on the network. DHCP relates to IP address, not MAC address.
Yes, I am aware that DHCP doesn't care about the MAC. What I do know is that most items that a connected to a network have a unique MAC, which is used to assign an IP address. Suppose that I wanted to access my Arduino from a network other than mine? Is that possible?
Yes, it's possible, you just need to expose the Arduino to that network. You can set a fixed IP address by specifying it as an argument to Ethernet.begin(). If you don't specify an IP address then DHCP is used, which increases the memory usage of the Ethernet library. You also need to call Ethernet.maintain() periodically if you're using DHCP.