The square mile is a unit of area and is an imperial unit with the symbol mi².
One square mile is equal in area to a square with edges one mile in length.
The Romans established the square mile; computed through a system of feet, paces and marches. The square mile's existence still remains in some places; for example "The Square Mile" is a term often used to describe the very centre of London in the UK (which just so happens to measure 1mi²).
The barn is a unit of area and is used in particle physics with the symbol b.
It represents approximately the area of the cross-section of a uraneum nucleus, so is very small compared to most other recognised units of area.
It is also used to measure the interaction of a nucleus with an electric field gradient in nuclear quadrupole and magnetic resonance.