There are many different systems of natural units. But the most interesting and fundamental are Planck units. Quantum Mechanics entails there is a smallest possible unit of space or time. Smaller units are physically impossible because anything that attempts to get smaller or divide them only ends up blurring across them.
The smallest unit of space is roughly 1.6 x 10^-35 meters and the smallest unit of time is roughly 5.4 x 10^-44 seconds. When we replace all units of distance and time with these, many fundamental physical constants disappear--they become equal to exactly 1, and since anything multiplied by 1 is just itself, the constants can be dropped from all the laws of physics without consequence.
For example, in natural units, the speed of light, or c, equals 1. So Einstein's famous equation, E = mc^2, is actually E = m x 1^2, and since 1^2 = 1 and m x 1 = m, in actual fact, E = m. The speed of light drops from the equation. The constant is no longer needed. This means the speed of light (and other constants which also disappear, including the gravitational constant and Planck's constant) is simply a physical product of the quantization of space-time, such that it would have the same value in every possible universe.
These constants therefore cannot be finely tuned. They cannot be tuned at all.