How does one measure the Fourier components of a signal?
Is the signal split by many narrow band pass filters in paralell? Are these filters RLC circuits? Are there chips that contains many such filters in paralell? 


No, these are typically collected as time signal, and then transformed digitally (by a DFT implemented by an FFT) to discrete Fourier domain for processing reasons. 

In the sense that yes, there's many hardware implementations of FFTs, and the FFT can be considered as polyphase boxcar filterbank, there's chips containing a filterbanks – but they're processing a digitized time signal, not an analogue one.

(The reasons being pulse compression, mainly, i.e. you need to do a huge correlation with the known signal to even see your signal from the noise – there isn't exactly much signal coming back to the satellite after being radiated from lower earth orbit to ground, through the ground, scattered there, and back. Correllation is very computationally intense, as it goes with the square of the length of the signal, but with processing in frequency domain, you can significantly reduce that amount, and bonus, the data might become way easier to compress, store and transmit to the ground station.)

(By the way, I wrote a top-of-my-head list of devices that do a Fourier transform during operation here, maybe you'll find it interesting.)

Generally, anything that can be done after  digitization digitally or before in analogue is usually done digitally — unless doing it in analogue reduces the digitization effort by magnitudes. Analogue electronics mostly have undesirable features for signal process – tolerances, non-linear in/out relationship that is hard to model, non-linear phases, temperature dependency, …

So, if you're in the business of processing signals, you typically want your signals to be digitized at the quality necessary, and afterwards do the math you want to do with the digital signal (which is actually numbers) instead of approximating the math you want to do with analogue components. You'll find that the same