At 1 bar, the boiling water is 372.78 K. At this temperature and pressure, the density of liquid water is 958.66 kg/m³ and that of gaseous water is 0.59021 kg/m³.

So the molar volume of liquid water is 0.0000188 m³/mol and that of gaseous water is 0.03053150574 m³/mol.

Assuming that water molecule excludes other water molecules from a cubic region centered on itself, how to estimate the average distance between nearest-neighbor water molecules in the liquid and in the gaseous water?

Molar volume of liquid water is given so from here we can work out the volume of 1 molecule and that is the space occupied by 1 molecule in a liquid state. Means just divide the molar volume by Avogadro's number to know the volume occupied by one molecule which we are taking as a cubic region.
Now two molecules in the liquid are separated in the form of cubic regions and molecules are exactly in the centre of these two cubes. If we calculate the distance of separation . that distance is the side of the cube.We have already calculated the volume of the cube by calculating the volume occupied by one molecule.Now just  take the cube root of that volume to know the side of the cube and distance between two molecules of water in the liquid state
Repeat the same thing with a density of gaseous water and molar volume of gaseous water from the molar volume. Calculate the volume of 1 molecule of gaseous water that is the volume occupied in the form of the cubic region. Calculate the distance between two gaseous molecules in the cubic region which is just the length of the cube. Take the cube root of the space occupied by one molecule to know the distance between two gaseous molecules of water.