Strain is the measure of deformation of a body relative to its original size. For direct (normal) strain it is defined as the change in length divided by the original length:

ε = δL / L

Here δL (the change in length) and L (the original length) are both lengths, carrying the same unit (for example mm or m). When one length is divided by another, the units cancel completely, so strain comes out as a pure number. This makes strain a dimensionless quantity — its dimensional formula is M⁰L⁰T⁰. It is sometimes written with helper labels such as mm/mm or m/m, or expressed as a percentage or in microstrain (με), but these are only conveniences; fundamentally the ratio equals a unitless number.

Therefore the correct description is that strain is dimensionless.

• It has units of length is wrong: although δL is a length, dividing it by the original length removes the length dimension. It is the elongation δL, not the strain, that has units of length.
• It has units of force per area describes stress (N/m² or Pa), which is force divided by area, not the deformation ratio. Stress and strain are related through the modulus of elasticity but are different quantities.
• It has units of volume has no basis; a ratio of two lengths can never produce a volume (which would need length cubed).

Hence strain, being a length-to-length ratio, is dimensionless.