We use the gravitational constant in this form because in Newtonian physics, it makes the most sense: it is the conversion factor between force (mass times acceleration) and the product of the "inverse square law" (m_1m_2/r^2) which has units of mass squared divided by length squared.
However, in relativistic physics, arguably a more natural choice would be to use G/c^2, which has units of length divided by mass. In other words, it is just a conversion factor between our preferred units of length and mass, just as is a conversion factor between our preferred units of length and time. We are allowed to free ourselves from convention and choose different units, in which = = 1. In such units, everything (the length of a ruler, an interval of time, the mass of an object) would be measured using the same fundamental unit, e.g., (if we so choose) the meter.
Curiously, Planck's constant is not like or , it is not a conversion factor. It is an "elementary unit of action" (here, "action" refers to the meaning of the word in Lagrangian physics.) By choosing = = 1, we find that Planck's constant defines a unit of area. Or, if you wish, its square root defines a unit of length, the famous Planck length. Then, using and as conversion factors, we can find the corresponding "natural" units for time and mass, and also derive natural units for, e.g., energy.
What are the units of the gravitational constant?
The gravitational constant, denoted G, is an empirical physical constant involved in the calculation of the gravitational attraction between objects with mass. 6.67x10^-11 N m^2/kg^2 (newton square meters per kilogram squared).