Translation:My hand hurts.
The plural is often contextual in japanese, couldn't "My hands hurt" also be valid?
No, generally we don't say "手たち".
When we want to emphasize "hands" as both left and right hands but not "hand", we say "両手（りょう て）".
手 here can mean "arm(s)" too, yeah? Just tried, "My arm hurts" and got marked as incorrect. Checked a dictionary and got these two definitions:
The upper limbs of the human body. The part extending from the left and right shoulders of the upper torso. The part from the shoulder joints to the tip of the fingers.
The part [of that] starting at the wrists. Also used not of this entire section, but vaguely of the fingers, palms etc.