HP 34C turning on but Error 0, zeros, frozen

[jebem]

From the published pictures, I see signs of battery leakage on the battery contacts, including the two PCB pins where the external charger is connected.

Assuming that you did not applied energy to the calculator using the external charger while having defective or absent batteries installed (that action can cause damage to the calculator as the charger is not a regulated power adapter), then I would use a ohmmeter to check for zero ohms (0.0 to 0.1 ohm max.) between the battery contacts and the PCB power supply side.

Or on a positive way, apply 2.4V to the battery contacts and use a voltmeter to read the voltage at the PCB power supply to be sure no voltage drop is occurring.

As the plastic flat cable copper traces are crimped/riveted to the battery contacts, there is a chance of bad continuity there.

When in doubt, and for testing purposes, I would use thin wires soldered between the battery contacts and the power supply, to make sure the calculator is getting the correct power voltage.

Note:
I would not apply heat or solder to the battery contacts rivets, as this will destroy the electrical contact and may even melt the plastic flat cable, making things worse.
If a bad contact is found, the quick and dirt fix is to use thin wire (wire wrapping wire) soldered to the battery contact.
A more professional, time consuming, and risky method would require to remove the metal rivet, clean, and apply a new rivet.