Use custom values as units? Does Blender have any way of using custom units? Say I create a "unit" called depth and set it to equal 5mm. I set an object to have the dimensions (1 depth, 5 depth, 3 depth) which would equal (5mm, 25mm, 15mm). Then I change depth to equal 6mm. The object dimensions would change to (6mm, 30mm 18mm). Does blender have anything like that?
Blender only has 1 unitless unit: the Blender Unit.
You can't create any others, but that doesn't matter, since it does exactly what you want.

At first your BU equals 5mm. Enter 0.005 in the Unit Scale checkbox.



Switching between metric and None will you either Dimensions of (1, 5, 3) or (5mm, 25mm, 15mm).

Changing the translating to 1 BU = 6mm seems pretty straightforward. Simply entering the new value 0.006 produces dimensions of (6mm, 30mm 18mm).