Electronic – How long will it take to completely run out of power for two A23 batties(12volt each) on a serial circuit with 6 leds

Your hunch that batteries have a current limitation is correct. In general, it's hard to tell the current rating [A] from capacity [A·h]. You have to look it up in the datasheet. A lot depends on the design of the battery.

For example: coin cells with 500mAh capacity may have only 3mA max current.
Another (opposite) example: automotive starter battery with 40Ah capacity may have 500A max current.

Lead-acid batteries are interesting in this respect, because there are two distinct types.

1. Starter lead-acid batteries are designed specifically to deliver high peak current for a short period of time. Deep discharge, however, dramatically shortens the life of a starter battery. So, it's not suited for routine operation at high depths of discharge. Your typical starter battery in the automobile works at very shallow depth of discharge.
2. Deep cycle lead-acid batteries are designed (as name suggests) to discharge further. But they can not provide as much instantaneous current.
   Here's an example datasheet for a deep cycle battery. Have a look at the nominal capacity on p.1. Notice that capacity depends on discharge current (i.e. the rate of discharge).

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Depth of Discharge    Starter Battery    Deep-cycle Battery

100%                  12–15 cycles       150–200 cycles

50%                   100–120 cycles     400–500 cycles

30%                   130–150 cycles     1,000 and more cycles

(Source.)

p.s. If you want to read-up, here's an excellent web site on batteries - Battery University.

One single large battery will be easier to deal with, and will also make it easier to switch all the lights on and off.

To determine how big a battery you need, do the math. Add up the current requirement of all the lights. That's the current the battery must deliver. Now divide the battery capacity by that current to get a time. That's the time, in theory, that the battery can deliver your current for. However, for off the shelf 12V lead-acid car batteries, you want to avoid running them very low because that will cause permanent damage. Either get a larger "car" battery, or a lead-acid battery specifically designed to be able to be drained to nearly empty. Such batteries are called deep cycle, or are sometimes sold as marine batteries.

For example, let's say all the lights together draw 1.3 A (just picking a number out of the air). A 50 Ah car battery can supposedly run the lights for (50 Ah)/(1.3 A) = 38 hours. However, let's consider only half that actually usable for a consumer level car battery, so 19 hours. You'd need to get a bunch of such batteries to last the 112 hours you want. In fact, (112 h)/(19 h) = 6, which is how many 50 Ah "car" batteries it would take.

Since you say you only need to run the lights 9:00-17:00 each day, a better solution is to use a single battery and charge it overnight. Presumably the gallery or whatever is closed during the off hours, so it doesn't matter how it looks with a cord running to a outlet during this time.