Electric water heater? A ground fault in the heater is one possibility. Turn off the main and see if the meter moves. If you have a well with water pump, there is a check valve to keep the water from going back into the well when the pump is not running. Called a foot valve with jet pump, check valve with submersible pump. If this valve is leaking, the pump will run much more often. Shut off the electricity to the pump and see if the pressure gauge reading drops while no water is being used. Over time, the extra cost for electricity will cost far more than having the problem professionally diagnosed and corrected.

I've been thinking about the water heater. If I turn off the power to the water heater, why would the meter move? And this house doesn't have a "main."

City water.

"And this house doesn't have a "main." "

There is either a main breaker at the first panel, or a split bus panel that has up to six 'main' breakers at the top of the panel.

The 240 V loads are on the upper breakers and also a 2-pole that feeds the lower portion of the panel.

OK, see if the meter does NOT move. I suggest hiring a professional.

The main panel is an old (late 50's) FPE and there's two sub-panels added (1) late 60's and (2) mid-80's. I'm pretty sure that sub-panel (1) isn't served by a breaker. So...obviously I could turn everything off in that sub-panel.

What I'm confused about is the suggestion of a ground fault in the water heater. Why would turning off the "main" (just to use the term) help me identify that? Shouldn't I be looking for continuity from an element to ground?

Or is turning off the "main" helping me identify another problem?

The basic strategy is to use the process of elimination to find the problem. You are talking about a pretty large power draw, so the most likely suspects are things with heating elements or motors. Chances are, one of those things is either running all the time or leaking power somehow.

So you start from the beginning. Turn everything off. Cut all the breakers. One by one, cut the breakers back on and watch the meter. You should be able to find the culprit that way. Once you figure out "what" is using so much power, then you still need to figure out "why" it is happening.

When electrical consumption increases unexpectedly, it could be one of many causes. Here are only a few.

A new appliance that uses more energy than you thought it would. This time of year, a new space heater wouldn't be unusual, for example.

Very cold weather and/or a higher thermostat setting. Furnaces use electricity, too.

An appliance accidentally left on in an unused room - a flatiron or space heater, for example.

Freezer door left open, or failed thermostat.

Other automatic device with a stuck switch. This could include sewage ejectors or aerators, sump pumps, and water pumps.

Electric water heater faulting to the water and from there to ground through a fractured element.

Damaged underground wiring - look for areas where the snow melts.

I don't see much use in opening the main, except maybe to be sure there's no leakage current ahead of the main panel.

First, turn off and/or unplug everything that you possibly can. If the electric meter on the outside of the house is still turning, open all the branch circuit breakers and see if it stops. Turn them on one by one and look for something that starts the meter turning. When you find it, it's time to go hunting on that circuit to determine whether that's a normal load or not.

It's a hunting expedition, but not that difficult or technical until you get into the actual repair. It just requires time and patience.