In some sense, you might consider it to be a crude one. If you configure your computer to boot directly into full screen emacs, never leave it, and do your damndest to interact only with in-process elisp, you might get some feel for what such a machine could be like.
It will also be a frustrating experience, since emacs was never really intended to be used in such a way. There’s going to be a lot of friction and limitations. Still, it’s a fun exercise for the right sort of person :)
Emacs predates Lisp machines, but a version of Emacs was written in Lisp and used there (EINE[1]). The point of a Lisp machine was that it ran Lisp natively or, at least, closer to native than, say, a 8086 DOS box.
EINE was the second Emacs (the first was TECO EMACS, an Emacs written in TECO macros) and EINE was the first one to be (completely) written in Lisp, on a GUI-based system.
(There is a lot I dislike about Emacs Lisp; there is a lot of power in it though. :)