I've got a junction box with some 1953 sheathed cable feeding it. Looks
like asbestos fiber sheath with silver spray paint on the exterior. Only 2
wires, both black. Measures 240V rms.

Leaving the junction box are some armored cables containing 2 wires each,
one black, one white. They also put out 240V rms. There are no neutral
wires or ground wires that I can see. The armored cable is well grounded
however.

I had expected that the black/white pairs were 120V, but they are not.

How bad an idea would it be to choose either the black or white wire and
connect it to the hot side of a 120V outlet, and then connect the neutral
side of the outlet to the armored cable?

And yes, I will call an electrician if I start to feel dangerous.