Disclaimer, none of this should be regarded as proof since it's all theory.
The 4th Dimension
The 4th dimension is by humans incomprehensible, the same way that the 3rd dimension is incomprehensible for a 2D entity and the 2nd dimension is incomprehensible for a 1D entity.
A 2D entity simply can't view its existence from a 3D perspective. Just like we are incapable of imagining viewing something from the 4th dimension, quite similarly how we cannot imagine a color outside our spectrum simply because it's outside of our perceived reality.
A 1D object simply has one dimension, "forward" and "backward" along a line. (so to speak)
A 2D object has two dimensions, "forward", "backward" and "side-to-side".
A 3D object has the dimensions of a 2D object with an added "up" and "down",
In short, 1D has one axis, 2D has two and 3D has three.
Now, some argue that the 4th dimension is "time" but that does not quite represent the entire picture.
We, as "3-dimensional" beings, comprehend the world and our universe as infinite 2D "projections" so to speak. In essence, we are viewing an infinite number of 2D planes. The same way a 2D being would observe their universe as an infinite number of "lines" of the first dimension. (this is not a 100% correct statement, but I hope you understand what I mean.)
So, A 4-dimensional being would be viewing their universe as an infinite number of 3-dimensional "projections". An "angle" or perspective we are simply unable to comprehend, the same way a 2D being is incapable of comprehending the angle a 3D being would observe it.
Thus, the 4th dimension would be an infinite number of 3D "instances" or projections. Now, some argue that this is what "time" means; "An infinite number of 3D projections over time". In other words, moving through a 3D space and through time.
However, for us to comprehend this fourth dimension, the fourth dimension needs to be presented in a "3D" format.
The interstellar movie had a pretty good take on this, as can be seen here:
Essentially, the fourth dimension would be presented as infinite instances in 3 dimensions; "up, down, forward, backward, left, right", that we would be able to traverse.
Note that these 3D "instances" should actually exist in the exact same place at the exact same time.
But that would mean that we would perceive it as only one instance, the one we are currently in.
In regards to OP's question:
If this theory is correct, your 4D cube of space would not "fill up" or suck water into it. It would probably merely be an infinite number of instances of the same water. (through time? maybe? nobody knows)
What this would mean that if the 4D "cube" was 1x1x1 meters in 3D size, the water inside it would be an infinite number of 1x1x1 meter instances of the same 3D water. In essence, an infinite number of the same 1 cubic meter of water.