I recently just ran a 4 conductor 240 volt line out to my shed. I chose to run the extra conductor just in case, but I do not currently need it for my application. My question is, can I add a 120V receptacle using the extra neutral line as long as I tap into the same ground as the 240V is using, or possibly another grounded outlet in the room? If not, then how do appliances make use of the fourth neutral to power clocks and things safely?
You didn't say what you're doing with the other side. It depends an awful lot on that.
If the circuit breaker in the main panel is 20 amps or less, and the existing load is 120V (i.e. between hot and neutral), you can make it a multi-wire branch circuit which is basically what you're talking about. (By "multi-wire" they mean multi-hot.)
Why 120V only? You can't do an MWBC if there are any 240V loads on the circuit.
Why 20A or less? Nothing about an MWBC requires that. But a different rule requires outlets have the same rating as the breaker. (Except 15A outlets are allowed on 20A circuits). And disallows ordinary wired lighting on >20A breakers. So if the breaker is 30A, you can't use common outlets or lighting.
If you do MWBC, it's best to feed it with a 2-pole breaker, which, by definition, takes 2 full spaces in the panel -- never, never, never a duplex/tandem breaker that crams 2 breakers into one space.
If the above rules don't allow you to do MWBC, then a sub-panel will handle it.
You cannot simply tap the neutral to put an outlet in -- however, since you have 4 wires running there, you can install a subpanel at the shed and feed the outlet from a 120V circuit on the subpanel. (You will have to drive a ground rod there and tie it to the subpanel ground, and of course make sure that the ground and neutral busses are separated.)
The 4th conductor is not "extra" (unless you actually have 4 + ground.)
Line1, Line2, Neutral (grounded), Ground (grounding).
Ground and Neutral isolated from each other, since this is a sub-panel type feed (whether or not you have a sub-panel installed - a sub-panel is the easy way to get what you want, though.) Ground connected to a ground rod (or two) and/or UFER ground if you have concrete foundation at shed.