How can I use my high impedance crystal radio headphones in regular nowadays appliances like mp3 players, cellphones and computers?
Have you tried connecting them straight to the headphone port? Depending on the sensitivity of the headphones, they may actually deliver acceptable volume.

If not, you'll need something to raise the voltage. You can do this in so many different ways. One method is to use something with a pro audio level line output - like a mixer. A balanced pro audio line out delivers a signal up to 60Vp-p which should be enough for the headphones.

Another method is to use a small audio transformer. The turns ratio of the transformer depends on the sensitivity of the headphones. You want something with a low impedance primary (8 to 32 ohms) and a high impedance secondary (several kohms or tens of kohms). Normally, good quality audio transformers are expensive. But considering the fidelity of crystal headphones won't be very high, you can just use a cheap transformer.

A third method is to connect the headphones to the output of a normal audio power amplifier. You may be able to connect them to the headphone port, but certainly they will work on the speaker terminals. The more powerful the amplifier the better, because you need a decent voltage. You're not actually delivering any significant power to the headphones, but you do need the voltage that a high power amplifier delivers.

The fourth method is to build your own amplifier around an opamp (5532 or TL072 are good choices). You will need to use a suitable power supply, probably +/- 12V or more. A PC power supply isn't a good choice because they're very low quality and noisy. You'll probably need to build your own supply. You might be able to use two or three 9V batteries. The gain required of the amplifier depends on how high the signal level from your audio devices is, and the sensitivity of the headphones. You might use a pot to adjust the gain, as a starting point I'd suggest you'd need an amplification factor of around 6 times (36dB gain).