Building a space elevator involves getting a substantial amount of stuff into orbit, i.e. the cable and counterweight, so this sounds a bit like pulling yourself up by your bootstraps.

Maybe you could launch materials and construction robots for an elevator using a mass driver then use the elevator for passengers, I dunno. Still need some non-rocket way to get things up there.

True, though in most fictional accounts the construction mostly happens out of the gravity well - e.g. move an asteroid or iceball into orbit as a counterweight, and start building down from it (and up from the surface).

And how do you get your spacecraft up out of the gravity well and out to the asteroid, so you can move it into orbit and start recycling it into elevator? It sounds suspiciously like you're solving the bootstrap problem by tying strings to your bootstraps and pulling those instead.