The first thing is that strictly speaking, there is not *the* natural unit system, it depends on the problem you are interested in. For example, if you are interested in atoms, the typical mass is that of the electron, so you will likely be interested in masses as multiples of $m_e$. Then, interactions are Coulomb and you will want to express charges as multiples of the electron charge $e$. Finally, quantum mechanics is your relevant framework, so it is natural to express actions in multiples of $\hbar$. Then a quick calculation shows that this unit system of setting $m_e=e=\hbar=1$ implies that distances are dimensionless and the distance $r=1$ happens to be the Bohr radius that sets the natural scale for the size of atoms. Naturalness here lets you guess the size of an atom from just identifying the electron mass, the electric charge and quantum mechanics to be the relevant ingredients.

When you are doing high energy particle physics quantum physics and special relativity are relevant and thus it is convenient to use units in which $\hbar=c=1$ which is Bee's example. In this unit system, masses and energy have inverse units of length.

If you are a classical relativist contemplating solutions of Einstein's equations, then quantum mechanics (and thus $\hbar$) does not concern you but Newton's constant $G$ does. These people thus use units with $c=G=1$. Confusingly, in this unit system, masses have units of length (and not inverse length as above). In particular, the length scale of a black hole with mass M, the Schwarzschild radius is $R=2M$ (the 2 being there to spice up life a bit). So you have to be a bit careful when you convert energies to lengths, you have to identify if you are in a quantum field theory or in a classical gravity situation.

My other remark is that it is conventional how many independent units you have. Many people think, that in mechanics you need three (e.g. length, mass and time, meters, kilograms and seconds in the SI system) and a fourth if you include thermodynamics (like temperature measured in Kelvins) and a fifth if there is electromagnetism (like charge or alternatively current, Amperes in SI). But these numbers are just what we are used to. This number can change when we change our understanding of a relation from "physical law" to "conversion factor". The price is a dimensionful constant: In the SI system, it is a law that in equipartition of energy $E=\frac 12k_bT$ and Coulombs law equates a mechanical force to an electrostatic expression via $F=\frac{qQ} 1{4\pi\epsilon_0r}$ and it is a law that light moves at a speed $c=s/t$.

But alternatively, we could use these laws to define what we actually *mean* by Temperature (then measured in units of energy), charge (effectively setting $4\pi\epsilon_0$ to unity and thereby expressing charge in mechanical units) and length (expressing a distance by the time light need to traverse it). This eliminates a law and a unit. What remains of the law is only the fact that one can do that without reference to circumstances, that a distance from here to Paris does not depend for example on the time of the year (and thus on the direction of the velocity of the earth on its orbit around the sun and thus potentially relative to the ether). If the speed of light would not be constant and we would try to measure distances by the time it takes light to traverse them then distances would suddenly vary when we would say that the speed of light varies.

There is even an example that you can increase the number of units to more than what we are used to (although a bit artificial): It is not god given what kinds of things we consider 'of the same type' and thus possible to be measured in the same units. We are used to measuring all distances in the same unit (like for example meters) or derived units like kilometers or feet (with a fixed numerical conversion factor). But in nautical situations it is common to treat horizontal distance to be entirely different from vertical distances. Horizontal distances like the way to the next island you would measure in nautical miles while vertical distances (like the depth of water) you measure in fathoms. It is then a natural law that the ratio between a given depth and a given horizontal distance is constant over time and there is dimensionful constant (fathoms per mile) of nature that allows to compute a horizontal distance from a depth.