The Planck mass is mite-sized and a billion times Planck mass is 48 pounds—roughly that of a grown dog like a husky or malamute. The NIST website gives the Planck mass as 21.767 microgram. The estimated uncertainty (given as 7.5×10-4) amounts to 1/13 of a percent.
Therefore the standard value (actually the CODATA recommended value) multiplied by a billion is 21.767 kilograms with uncertainty 1/13 percent. This is equal to 48 pounds within experimental error
So one can picture the natural unit of mass—something implicit in the proportions of light and gravity to be found anywhere in the universe these things are examined closely—as a flea or mite, the size of the period at the end of a printed sentence, and a billionth of the mass of a dog.
The natural mass unit is simply (h-bar c/G)½.
In case that doesn't come through: SQRT ( h-bar c/G).
h-bar, c, and G are not going away. They'll be here when our species has turned into something else or vanished. The Planck mass is just the mass arising from those things and it will not go away. It's the mass unit intrinsic to light and gravity and constitutes one of the universal natural units.
One way to think of why the Planck mass formula is what it is:
Every quantum of light carries a certain quantity of energy and has an (angular) wavelength. Because of a tradeoff the energy×length product is the same for all quanta.
More energetic wavelets of light have shorter wavelength, so the product is always the same, namely h-bar×c.
But gravity associates to any mass a certain energy×length product as well. This product is the binding energy of a pair of bodies each with that mass multiplied by their separation. Again there is a tradeoff: increasing the separation decreases the binding energy (the work that would be needed to drag the two so far apart that their attraction for each other is negligible, in effect to infinity).
The Planck mass is that mass to which gravity associates, in this way, the energy×distance product which is present in all light—intrinsic to each individual quantum.
To say the same thing algebraically, G times the square of Planck mass is h-bar×c. So the mass itself is the square root of h-bar×c/G.
A second way to understand the formula for the mass is via the Planck force c4/G. This force is obvious everywhere in the universe since it connects the curvature of a region to the concentation of mass-energy there. Indeed Planck force is the central constant in one of the principal equations of General Rel. So that makes Planck power pretty obvious too, namely c5/G.
But then h-bar connects power to the square of frequency. So Planck frequency is just SQRT(c5/Gh-bar) and h-bar times the frequency is Planck energy SQRT( h-bar c5/G)
Then one divides the natural unit of energy by c2 to get the mass.