Earlier quoted context omitted.
The space elevator even on Earth in GEO is a hopeless, fool's dream.
A lunar space elevator is feasible with today's materials (just a whole lot of them, way too much for current technology to transport to the moon). Here's a nice talk by Markus Landgraf, a mission planner at ESA, at 33c3: https://media.ccc.de/v/33c3-8407-an_elevator_to_the_moon_and...
Just because something ticks the checkbox of being theoretically possible doesn't mean it's practical or feasible to build.
Maybe in a few hundred years we'll have the means to build something on that scale. It still takes $10k to put 1kg in LEO, imagine the cost of transferring to the Lagrange points.