What is the acceleration due to earth's gravity at a distance from the center of the earth equal to the orbital radius of the moon?

The gravitational force exerted by the Earth on an object, and thus the acceleration due to gravity of the object, is inversely proportional to the square of the distance between the center of the Earth and the object. 

At the Earth's surface, with mean radius 6380 km, g = 9.81 m/s². 

Mean distance of Moon from Earth is 384000 km. (The moon takes quite an elliptical path, and its orbital distance varies from about 363000 km to 407000 km.) 

So acceleration due to Earth's gravity at mean lunar orbit = 9.81 * (6380/384000)² = 0.0027 m/s², to 2 significant figures.