I've read that it's nearly impossible to take a proton from an element. But if it's "nearly" impossible then it is possible to some degree. If this has happened, what is exactly the process of taking an atom apart. Likewise, would this be possible for electrons and neutrons as well?
Taking out electrons is the easiest of all: this happens in a cathode ray tube TV all the time. A filament in vacuum is heated such that the electrons have enough energy to leave the filament. Applying a voltage of the order of 10 kilovolts will accelerate them towards the screen (and prevent them from going back to the filament).
To give an order of magnitude, the binding energy of an electron (in its lowest state) in a hydrogen atom is 13.6 electronvolts.
Taking out protons and neutrons of a multi-nucleon nucleus requires more 'effort' (energy): the so-called binding energy (energy to be 'surmounted' to liberate a proton or neutron from a nucleus) is typically of the order of Mega-electronvolts, see for example the graph of nuclear binding energy per nucleon here.
In practice, to 'excite' a proton or neutron enough, i.e. inject the binding energy or more, one can shoot high energy protons or photons at the nucleus. However, most likely all kinds of reactions other than the desired one will happen at the same time, the efficiency of creating the desired nucleus with one proton or neutron less may be quite low (and one needs some filtering mechanism like a Wien Filter behind the target).
In practice, there are often more efficient ways of producing a given isotope, especially when the isotope in question occurs naturally (e.g. Deuterium).