Code on capacitors

I'm a little surprised that the book tells you nothing about how to actually make the inductors. For AM, you generally make what's called a 'loopstick' antenna, like so:

(diagram from quazar31)

The idea is that you wind a primary and a secondary coil such that the flux in one affects the flux in the other, and vice versa, hence it acts as a transformer.

Whenever you wind a coil, the inductance is proportional to the number of turns and to what's called the permittivity of the substance inside the coil. Ferrite has a much larger permittivity than air, which is why it's frequently used in inductors. See the Amateur Radio SE for a more detailed discussion of ferrite parameters. Of course, you can also make air coils, but you will need many more turns.

A resource like the ARRL handbook will give you rules of thumb for getting the correct inductance. Having an inductance meter (or a function generator and 'scope) helps for double-checking.

It looks like your fried cap is connected right next to the input terminals of the +12 V rail and GND. Here are the steps I would try to get the drive running again:

1. Remove the cap. Power the hard drive, chances are it will work anyway because the cap just acts as a small filter for the incoming +12 V rail and the slightly increased ripple voltage on this rail will likely be tolerated by the rest of the hard drive. This test will also tell you if anything else has failed along with the cap or if you're lucky and the cap is the only bad part.

2. If step 1 was unsuccessful, you can put pretty much any cap with at least the same capacitance and at least the same voltage in place of the fried cap. Maybe the drive will work now? If you just need the drive to recover the data, any half-decent electrolytic (including tantalum) capacitor will likely do the job long enough until you finally scrap the drive. Don't worry too much about the ESR. As the cap appears to be just in parallel to the 12 V input, it's not worth counting every milliohm. However, your're of course right: The smaller the ESR, the better.

3. However, by the looks of it, the drive is so old that you value not only the data but the cool vintage hardware itself. Here's what I would do for a good, permanent repair: Sizes of SMD tantalum capacitors are standardized. Yours appears to be one of the bigger ones (C?, D?). Try finding a replacement cap with the same size and capacitance, and you're very right about the voltage - 25 V or more is a very good idea. There is a huge derating to be considered when using tantalum capacitors, and using a cap with twice the voltage it is subjected to in your application is not a bad idea at all. One last hint: While aluminium electrolytic caps have the mark ("bar") on the negative end, tantalum capacitors are marked on the "+" end!

4. As we're aleready talking about derating and fried tantalum caps: Besides (even very short) voltage spikes causing tantalum caps to fail catastrophically, they are also very sensitive to current spikes (as they will appear when you hot-plug the power connector into the drive). If you have the space, using an aluminium electrolytic cap as a replacement will be more robust. If you want to go for a "good as original" vintage repair, try to stay away from hot-plugging your drive.

Example random size chart (it gets more confusing when you consider not only the length and the width, but also the height):

Related: https://electronics.stackexchange.com/a/80458/930