I want to run a couple of 110 volt plugs from a box that has 220 volts. Both the black and white wires are 110 volt and there is only a ground wire with them. Can I tie into one of the hot wires and just connect the neutral wire to ground?
That would be a National Electrical Code violation and a dangerous practice.
That would energize the the ground wire and someone could be electrocuted from a wire they can normally expect to be safe.
If you cannot pull new cable with a proper neutral AND ground conductor separately, then the only legal way to attain 120V would be to use a 240-120V transformer and power the transformer from the 240V circuit, then let it create a new 120V circuit from it with a legal Neutral that is now bonded to ground at the transformer. This is called a "separately derived source". That transformer would also need to have proper primary and secondary circuit protection based on the size / capacity of 120V that you need. Trust me, it would likely be cheaper to rip open the walls and run a new cable...
Neutral is not ground, and you must never misuse neutral for ground or vice versa. It can cause a serious safety hazard.
Most 240V circuits are dedicated to a single machine, device or receptacle.
If this circuit serves only this location, or if you are able to convert all its locations to 120V, you can convert this circuit to 120V.
Change how it's punched down in the service panel, so instead of white and black going to the two sides of a 240V breaker, the white goes to neutral. (the black can still go to 1/2 of a 240V breaker, so you don't even need to move it). Now change each receptacle to the appropriate type.
Remember that if your circuit breaker is 30A, all its receptacles must also be 30A. If you need to plug 15-20A plugs into that, check out a PDU or power distribution unit intended for computer servers. You could also downsize your breaker to 20A, as the breaker size decides circuit size. Don't upsize breakers without fully understanding wire size and the rules there.