24KHz isn't a particularly low frequency so the inductance of the antennas shouldn't need 'thousands' of turns of wire.
I have to confess this isn't my field of expertise either but it sounds like you have a simple VLF receiver, I'm not sure why two ferrite rods are needed though unless they are mounted at 90 degrees to each other to avoid signal nulling. Is it possible for you to post a picture of the unit to give us a better idea of what you are trying to emulate?
The basic operation of a VLF receiver is to use tuned circuits at the desired frequency, then an amplifier to increase the signal level, followed by some sort of measurement system. This could be a meter (analog or digital) or an acoustic output. Given that the original is around 50 years old, it should be possible to make a more efficient version now but some physical factors may restrict it's minimum size.
The number of turns on the rods is impossible to predict without knowing the type of Ferrite and the value of any parallel capacitances. The resonant frequency is "2 * pi * SQRT( L * C)" where C is probably a fixed capacitor connected across the coil, the frequency is the one you want to monitor, in your case 24KHz and L is the inductance of the coil. The value of L depends on the number of turns and by how much the Ferrite is increasing it. There are many different grades of Ferrite, each having different properties. With so many variables it's impossible to say how many turns should be on the coil. In general though, you can compensate for shorter Ferrite rods by increasing the number of turns or increasing the capacitance across it. The thickness of wire also influences the inductance but not anywhere near as much as the number of turns so with shorter rods if you use thinner wire you should be able to squeeze the extra turns on it. Alternatively, if there is space, you can wrap more than one layer of wire, making it fatter rather than longer.
Brian.