A while ago, I decided to try to make a zombie survival MMO. But, I've ran into a big problem in finding an engine. Basically, I need a good MMOFPS engine, preferably open-source (or easily moddable). My budget is $0, which is the main problem I'm having...

Anyone have any suggestions? Am I being overly ambitious? Should I just give up? Or what?

The Cube/Sauerbraten engine is open source, and I often see Reality Factory recommended as "free'n'easy"; appears to be open source as well but it advertises itself as a "toolkit for creating 3D action, adventure, role-playing, machinima and many other types of games", so it might require more work on your part.

Of course, if neither of those sound appealing you can try a feature search here for an engine that matches your requirements. Or if you already own a game such as Unreal or Quake you could attempt to mod it, quite a bit of documentation online to help do so.