What does it mean when both my breast hurt with sharp pain?

Probably cysts: It sounds like you have fibrocystic tissue that can fluctuate during a females normal monthly cycle. If your cycle is irregular it is more likely to give you retention cysts in the breasts that can them cause pains. Try to determine when the pains are worse and see if it coincides with a certain time of your monthly cycle. Another time that breasts tend to have pains is with pregnancy.