Type of Storage Battery

All batteries are combination of cells. So yes you can get one at 60 Volts DC. At 60 volts may not be a stocked size so you will have to have it made up. Or buy stock batteries and connect them in series. 5 twelve volt batteries will give you 60 volts. I do think they make an industrial 60 volt for forklifts.

Yes, accumulators using lead-acid (secondary) cells have been used for well over 100 years for telephone exchanges, the standard voltages being 60V in United States and 48 volts in Europe [24 cells x 2 volts = 48V].

Lead acid cells are reckoned to have a nominal voltage of 2 volts per cell, on discharge, however, it really depends on the discharge current, the state of charge and the particular design - the open circuit voltage is directly related to acid specific gravity, being about 2.05V for long-life stationary cells and 2.1 for motor car starter batteries. Discharge voltage is usually about 1.9 volts per cell at the end of a 10 or 20 hour rated discharge, but could be as low as 1 volt/cell for engine starting.

In lead acid cells, acid S.G. falls during discharge and can be used as indication of state of charge - in other types, like nickel-alkaline, the electrolyte itself is not altered by the chemical changes producing electricity.

There are many other types of accumulator cell e.g. Lithium, which has a much higher voltage per cell (up to 3.6 V), but one can always use cells in series to get a higher voltage.

There are even more types of cell which store electricity chemically, but cannot usefully be recharged electrically (primary cells). Unlike secondary cells, which (fundamentally) have to be charged before use, they deliver power immediately. The most used type is the Leclanche with carbon and zinc electrodes and the so-called Alkaline or manganese cell which is chemically the same.

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To make a summary of the two other fine answers already provided, the answer to the OP question is No and Yes.

No, the voltage produced by the chemistry of each storage cell cannot reach 60V

Yes, these cells can be series connected to achieve 60V.

Well, they're a lot more common than storage batteries that develop 60 V A.C.

Well, I don't think there is any 'Battery' that produces A.C. power, whatever be the voltage !

Precisely the point I was making!

Yes. I have five 12DVC ones, wired in series.