> Shure Stratoliner 708a... Does this just use a typical XLR?
Not at all. It needs a HIGH! impedance load, like a tube grid.
Wire it into a guitar amp as shown below. That "should" work. You may have to very-close-talk to hear yourself.
If no... use a capacitance meter at holes 2 and 3. Should be around 1500 pFd. If just a couple pFd, open it and look for the broken wire. Beware: the crystal is fragile.
OK. Get a 2-inch speaker and Radio Shed's 1KCT-8 transformer. Wedge the speaker into the Shure case; cardboard and foam may be useful. Wire the speaker to the 2-leg side of the transformer. On the other side, take the center pin and an end pin. (Bend-over the unused endpin so it won't touch anything.) These go to the two signal pins on an XLR. The mike case goes to the XLR ground pin.
It isn't "same as", not by a long shot. But it is lo-fi with a strong 4KC peak and narrow beamwidth in treble, just like the Shure used to be. And it interfaces just like an SM58, plain old 250 ohm dynamic mike. Goesinta any "normal" (modern) XLR mike input.