When wiring a 120V outlet from a 240V 3 wire service can you just connect the neutral white to the middle wire of the 240 and grab a ground from another source?
The 240 transformer that delivers power to your home has a "center tap", which gives 120 VAC to each side from the center and 240 from "hot" to "hot". It sounds like you're describing it correctly. Use the center tap and one of the hot lines to give you 120VAC, and there should be a ground bar inside the breaker panel that you would use to provide ground to the plug. The neutral and ground may or may not come from the same "source", depending on what you're trying to do. If you run a 240 feeder to a subpanel in a separate building you run two hots and a neutral and you put in a separate grounding rod to connect to the ISOLATED ground in the subpanel. The neutral and ground are not allowed to be connected together in that configuration. If you're running a 120 circuit instead, you run hot, neutral and ground together from the main panel to the subpanel.