Is it possible to write a 4000 word university essay in 2 days?

That's about 16 double-spaced pages.  It's certainly possible, but I wouldn't plan to do much of anything else during those 2 days.  If this just includes your writing time, then that's much more doable; but if you still need to look up information, organize your thoughts, etc. then you will have a very rough few days.

If you know what you need to say, have completed your research, and are ready to work, yes, it's possible.

That’s about 16 pages double-spaced. If your essay requires research sources, I don’t think it’s possible to write a decent paper.