I wanted to get my hair cut short.

Literal

I [topic-は] hair [object-を] short-[adverbial-く] cut-[have-done-てもらう]-want-[past-たい].

Complex auxiliary stack: 刈る (cut) + ~てもらう (have done by someone) + ~たい (want to) + past. 髪の毛 ('hair,' literally 'head's hair'). 短く is the adverbial form of 短い, used as a resultative ('so that it becomes short'). 刈る is specifically for cutting hair or grass.