The base SI units are in fact distance (m), time (s), temperature (K), amount of substance (moles), mass (kg), electric current (A), and luminous intensity (candela).
All other units are derived. From mark's list:
Mass and Length are base.
Height is a length, so also base, as is width.
Area is m x m, aka m^2
Space and volume are the same, and are m x m x m, m^3.
