Chattering contactor, what should voltage be when AC is on/off?
If the transformer does not have one leg grounded, then you could measure 24 volts at the contactor when the thermostat is not calling for cool. This is because you're measuring between one leg of the transformer (24 volts), and a floating wire (0 volts).  

When the thermostat calls for cool, you'll still measure 24 volts if the leads are not attached to the coil. Because you're measuring between each leg of the transformer.

If the leads are attached to the coil, and the thermostat is calling for cool. Then the circuit is complete, and you should measure 0 volts. Because both leads are now at the same potential.

If the coil is bad, or there's a loose connection or broken/damaged wire somewhere. You'll see the voltage "bounce" between 0 and 24 volts, but probably only if you're using an analog meter.

Check all the wiring for loose or broken connections. Make sure the contactor is the proper size, and that the coil is still good.