Pi does not have units because it is a number, not a measurement. However, pi is sometimes treated like a unit when referring to angle measurements because the actual unit, radians, is a length divided by length and is therefore dimensionless.
All "pure" numbers are defined as dimensionless quantities. These are the numbers that we use to count, including the irrational numbers like pi and e. Although some dimensionless quantities can have units (like radians), the units are only used when pure numbers are used to measure a quantity. Thus, it is possible to count to 100, but if discussing the length of a football field, it does not make any sense until a unit is added, for example, 100 yards.