My bet is these were oil-filled, glass plate capacitors. Such capacitors are an interleaved stack of thick glass plates and smaller brass sheets immersed in a bath of insulating oil. Alternate metal sheets are connected together to form the two electrodes of the capacitor and the connections are brought out the top. My experience is with a much smaller version with six or eight 1/4 inch glass plates about 8 by 15 inches dating from the early 2oth century. Several of the plates were cracked and needed to be replaced.
These capacitors were part of a very impressive old diathermy machine I was fortunate to obtain in high school. The coil on top of the 5 foot high apparatus would emit a hissing ball of crackling high-voltage, high-frequency electricity when energised. Although a Tesla coil was part of the apparatus, the spectacular coil on top was an "Odun Resonator," which had a lot in common with early radio transmitters. A fluorescent lamp held within six feet of the high-voltage terminal would light wirelessly.
The apparatus was driven by a 60 cycle AC transformer producing perhaps 20,000 volts on the secondary. The transformer discharged across a spark gap assembly consisting of multiple gaps which could be varied in spacing to control the discharge coltage. The spark gap excited a tank circuit consisting of two capacitors and a few turns of the primary high-voltage coil. The entire device probably resonated in upper HF or lower VHF region, perhaps in the vicinity of 30 megahertz. Radio frequency energy is perceived by the body as heat at these frequencies. Diathermy was a medical procedure which would produce deep heating in the tissues to treat aches, pains and anything else physicians could charge for. Think microwave oven on the end of a wand. I'm sure something magical was claimed for the process, but what you felt as a patient was warmth inside your joints, limbs or wherever.
As Tesla famously produced electrical discharges from towers which looked like lightning, his apparatus would have required multiple, equally large capacitors to deal with the voltages involved. The reason these large "tanks" have not been stolen over the years is probably because they weigh several tons each and contain nothing much that would be of salvage value except perhaps the oil, which would have been easy enough to drain off. Try salvaging large, thick glass plates without a crane. They would probably be considered not only awkward but worthless.
The riveted construction would have been typical of the time and the tanks were probably made from iron or steel plates, also used in boilers, tanks and ships. I'm sure the devices are described in one of Tesla's many patents. Given sufficient energy and time, I'm sure the mystery of the nature of the tanks could be solved. But I'm sure they weren't batteries. Given Tesla had them made, they had to be vats for insulating oil containing capacitors or transformers.
So is this a fact, a factoid, an opinion or a reminiscence? Have it as you would. The only part of my high-school apparatus that survives in my posession is the large, foot operated on-off switch. The switch is mounted on a slate plate and has an insulated actutor of something like Bakelite about six inches long. Oh, just read about it in the 1923 patent:
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