It's not theoretically impossible, by any means. After all, the Silbervogel that started things off used rockets, and the laws of physics work just as well now as they did then. The problem is weight, or more precisely mass. A rocket launch would have to use pure thrust to overcome gravity and gain altitude, which demands an immense amount of fuel. That fuel then needs more fuel to be burned to accelerate it until it gets used, and so on and so on. Unless you've got some kind of very energetic fuel in mind, the payload could only be a tiny fraction of the rocket, and at that point I doubt any remotely practical design could carry astronauts to the moon - at least, not if you wanted to get them back again!
I think it's far more practical to build your rocket in orbit. It would take a lot of flights, but spaceplanes could absolutely carry the parts of a rocket up to low-orbit for assembly. You'd need to figure out some way of keeping people alive up there for more than a few hours, but honestly you'd need that anyway if you're sending people off to the moon, so it's not wasted effort. Building and launching from orbit vastly reduces the fuel requirements, and drags the whole idea into the realm of the practical. Build it, fuel it up, and off it goes. Much easier.
The whole idea of a single giant rocket just doesn't make much sense to me. Like I said, it's not absolutely impossible, but it would be very difficult and I honestly can't see why you'd WANT to do it that way. Much better to go with some sort of incremental plan like I suggested.