i believe that everything that we see - being any form of compound, bound by electrostatic attraction and thus forming a solid, liquid or gas, will then be bound by those same limited bonds, and therefore develop an elastic material property
to extend beyond this field strength - breaks the bond , and therefore breaks the field
engineers typically refer to material properties as yield strength , youngs modulus, ...(etc), in materials that are worth considering as structural conduits for transferring forces
if the force is great enough, you've simpy broken your model - try again, it gets bigger and bigger until it works, much like church structures during the dark ages, buttress piers simply got bigger until the building actually stood up
if you also consider the gravitational attraction from earth to the moon, the element (unless weightless, also an impossibility) when you integrate over the distance of attraction from earth, and then the moon - would require a very very very heavy end at the moon side so as not to fall back to earth - dragging said astronaut with it...
then the rope/string/element would need to be positioned at the pole so the humanoid wouldn't have to run very fast, at least the astronaut fares well at this point - always facing earth from the same spot..
however, gravity is a b%$ch, the rope would be tugged into a parabolic catenary sag and not be straight any longer - so is now a cable reliant on multiple force systems to maintain equilibrium. ( the poles are not in line with the string connecting moon to earth, so there would be a lateral component on the cable for a considerable distance away from the surface of the earth, but not necessarily so from the moon side)
the tug would then resort to a wave force similar to a wave in the ocean, a wave on a string, etc, and thence travel as such to the moon... albeit a bit slower than the speed of light
so stop the earth spinning, and you have a chance