You have a lost neutral
... and the chassis of the dryer was electrified. You're lucky you didn't meet your maker.
That third wire is neutral, not ground. There is no ground.
Perhaps you are accustomed to welders, compressors, etc. On those, the three wires are hot, hot and safety ground.
However, in a dryer, the three wires are hot, hot and neutral. Several parts of the dryer are 120V equipment and require the neutral to function. This connection type does not have ground, which is why it's been outlawed for 30 years.
Your neutral wire is inoperative. So the timer, controls, thermostat and tumble motor cannot turn on, therefore the dryer can't run.
Even worse, the load sitting there, connected, between a phase and neutral, is sitting there like a low value resistor, "pulling the neutral over to hot voltage". That is why neutral reads 0V from one hot and 240V from the other.
And it's trying to kill you.
So what do they do to "ground the chassis" on these obsolete 3-wire connections? They bond the chassis to neutral, doing the very thing that is called "bootlegging" when it's done anywhere else.
The logic is that neutral wires rarely fail.
So, since neutral is the same voltage as a hot wire... and chassis is bootlegged to neutral... that means the dryer's chassis is also at the voltage of a hot wire. It's been sitting there waiting for you to touch the chassis and also something grounded!
And maybe you've felt some little bites - 99% of the time you'll get lucky. It's the 1% that will turn you into an obituary. And when they write it, they'll say the dryer circuit was miswired -- no, it was not. It was wired completely correctly (for 1989). Simply, the neutral wire broke or has a poor contact. That's all it takes for the setup to turn lethal... and that's why it's outlawed.
Convert the connections to 4-wire.
You can either retrofit a ground wire ... or, you can use the 4-wire socket without a ground wire, and drive it from a GFCI breaker in the panel. Label the outlet "GFCI Protected / No Equipment Ground".
Note that if your wiring is 3-wire black-WHITE-bare, that wiring was never legal and is not grandfathered and must be taken out of service. It can be re-tasked to drive a 120V circuit, such as the mandatory 20A/120V circuit laundry rooms are now required to have.