(one) only has one pair of hands
There is a limited amount of things one can do at once, so do not make unreasonable demands or have unrealistic expectations. You'll have to wait until I'm finished cooking dinner before I can come help you—I only have one pair of hands! She only has one pair of hands, so I don't know how she'll get that all done before the deadline.
I only have one pair of hands
1. There is a finite amount of things I can pick up, carry, or do with my hands. A: "Don't forget to carry the groceries in from the car." B: "Can you come out and help? I'm bringing the kids inside, and I only have one pair of hands."
2. I am too busy to take on any other work. The boss wants me to go over all the intern's work, on top of what I'm already trying to finish for Friday. I only have one pair of hands