The Delta V requirement to launch is about 14 km/s to low lunar orbit, per Wikipedia. That means that you would have to achieve a speed of 14 km/s in order to orbit the moon. Some of that will need to be done from space, but most of it could theoretically be achieved from the ground. So, what do you need to do to make that happen?
In World War II, the Germans developed an artillery shell that could travel at 1.67 km/s. It used 200 kg of powder, and fired a 106 kg shell. Let's just suppose that you could scale that upward infinitely (Not likely, but we'll just assume for a moment). Furthermore, let's assume a mass of 1000 kg for the ship (Likely would be higher). Given all of that, you would need 10 times as much to launch the ship the same speed, and about 72 times as much to launch the ship to lunar orbit. That would scale to about 14400 kg of powder, or about 16 tons of powder, much less than Jules Vern stated you would need. So, why don't we do that?
While in theory one could get to the moon like this, pure cannon thrust would not be sufficient to land on the moon, at least in a controlled manner. You would end up landing on the moon at the lunar escape velocity of 2.4 km/s, without a rocket to stop you. Furthermore, the gravity forces exerted on you would at launch would be enormous, artillery shell electronics have to be rated at 15000 Gs. Good luck getting a person to survive that. And also, the physics doesn't quite scale as I indicated here, but the numbers provide a good first order approximation.
The launch profile of a rocket is near best case for getting astronauts to Space, in terms of the amount of gravity. You really do need to be continually thrusting for some time. However, a rail gun could provide some of the velocity needed to orbit, if you plan carefully to make this happen.