I work for a Emergency response contractor. We are on call 24/7 which is not a bid deal but I am curious if I get called out are they required to pay a specific amount of hours. Is there such a thing as a 4 hour minimum?

Not in your state, no.

How about Kentucky where our corp headquarters are?

Still no, not that labor law cares where the company is headquartered. Labor law is a function of the state that the work is done in. For example CA has a reporting time rule, but even if your company was headquartered in CA, both OH and CA would be very clear that that CA rule would have no effect on work done in OH.