I haven't done the numbers (I'm a bit rusty with architectural structures), but I guarantee you a simple compression tower 250 miles high with a base diameter of 4 feet will crumple sideways under its own weight, even on the Moon.

A realistic design for a compressively-loaded steel structure Moon tower would actually look something like the Eiffel tower, and figuring out what the maximum height of such a structure would be on the Moon is beyond my simple skills.

The more straightforward method for building space elevators is to deal with purely tensile situations, where the structure is loaded in tension (from a counterweight in space). This needs a lot of work on materials for an Earth application, but from what I have read, current materials could be used for a Lunar version.