When we set $c=1$ and $\hbar=1$, why is energy still measured in $eV$? When we set $c=1$ and $\hbar=1$, we often see in particle physics that mass and energy are expressed in terms of $eV$.
This doesn't make sense to me. If we are choosing a new unit system where for example the units are $apple$ for length, $bear$ for time and $cat$ for mass such that speed of light $c=1 \frac{apple}{bear}$, shouldn't the mass in this unit system be expressed in terms of $cat$ and energy in terms of $cat \frac{apple^2}{bear^2} $? Why are we still using $eV$? It is not a valid unit in the new unit system.
I also thought that maybe setting $c=1$ and $\hbar=1$ is just a useful mathematical trick by letting $$3\times 10^8m=1s$$
$$6.6 \times 10^{-16} eV = 1 s^{-1}$$
so that $c$ and $\hbar$ disappear in our equations. In other words, we are not actually using another unit system but just doing a mathematical manipulation for ease of calculation. But this answer says that we are indeed using another unit system.
 A: You appear to have misunderstood the natural system of units. Most HEP courses cover it on the very first lecture on the subject (at least Feynman's did).
The system reorients [L], [M], [T] to [V],[S],[E], namely speed, action (or angular momentum) and energy, respectively, and then measures speed in units of c, action in units of  ℏ, and energy in units of  eV.
So your apple/bear/cat paradigm is flawed. c is 1 in units of c, and x=something in units of apple/bear. Nobody set c=1 in those units!
Likewise, ℏ is 1 in units of ℏ. As a result, all quantities are in units of $(\hbox{eV})^a~ \hbar^b~ c^d$, but, if you know their dimension, you may skip d and b and only measure them in units of energy, so eV.
You may then reinstate the units of speed and action by dimensional analysis, uniquely, since you know their dimension,  if you need to communicate the final result (a frequency/rate, etc...) to an engineer for a macroscopic measurement. But since the  contact points of HEP to the macroscopic world are limited, you very-very-very rarely need to do that, and you live a life of one unit.
Try an example. Can you see how , e.g., lengths are measured in units of
ℏc/eV; masses in eV/$c^2$; and lifetimes in ℏ/eV ?
