Which of the following measurements is most precise?

The correct option is 5.00 mm.

Precision of a measurement refers to the limit or resolution to which the physical quantity is measured. It is determined by the least count of the measuring instrument used. A measurement with a smaller least count (or resolution) is considered more precise.

Let's analyze the resolution/least count of each given measurement based on their units and decimal places:

1. For the measurement 5.00 mm:
The measurement is recorded up to two decimal places in millimeters (mm).
The resolution (least count) of this measurement is:
$$0.01\text{ mm}$$ which is equal to:
$$1\times {10}^{-5}\text{ m}$$

2. For the measurement 5.00 m:
The measurement is recorded up to two decimal places in meters (m).
The resolution (least count) of this measurement is:
$$0.01\text{ m}$$ which is equal to:
$$1\times {10}^{-2}\text{ m}$$

3. For the measurement 5.00 km:
The measurement is recorded up to two decimal places in kilometers (km).
The resolution (least count) of this measurement is:
$$0.01\text{ km}$$ which is equal to:
$$10\text{ m}$$

Comparing the resolution (least count) of all options, we find that 0.01 mm is the smallest value (highest resolution). Therefore, the measurement 5.00 mm is the most precise.