In book "from eternity to here" by <PERSON> he described distortion of spacetime near a black hole. also you can change the visualization to realize milder distortions. Before you go forward you should know about light cones in GR. it is explained in book but I assume you know it. He says:
[But a real black hole, ..... It is a true region of no return—once you enter, there is no possibility of leaving, no matter what echnological marvels you have at your disposal. That’s because general relativity, unlike Newtonian gravity or special relativity, allows spacetime to curve. At every event in spacetime we find light cones that divide space into the past, future, and places we can’t reach. But unlike in special relativity, the light cones are not fixed in a rigid alignment; they can tilt and stretch as spacetime curves under the influence of matter and energy. 43
In the vicinity of a massive object, light cones tilt toward the object, in accordance with the tendency of things to be pulled in by the gravitational field. A black hole is a region of spacetime where the light cones have tilted so much that you would have to move faster than the speed of light to escape. Despite the similarity of language, that’s an enormously stronger statement than “the escape velocity is larger than the speed of light.” The boundary defining the black hole, separating places where you still have a chance to escape from places where you are doomed to plunge ever inward, is the event horizon.
It would be wrong to think of the singularity as residing at the “center” of the black hole. If we look carefully at the representation of spacetime near a black hole shown in Figure 19, we see that the future light cones inside the event horizon keep tipping toward the singularity. But that light cone defines what the observer at that event would call “the future.” Like the Big Bang singularity in the past, the black hole singularity in the future is a moment of time, not a place in space. Once you are inside the event horizon, you have absolutely no choice but to continue on to the grim destiny of the singularity, because it lies ahead of you in time, not in some direction in space. ]
So if I understand it well (and I should tell I am not a physicist) I can say the rubber sheet visualization is good for describing the space distortion near a massive object (look at the light cone near singularity). Objects don't absorb the space instead they push space like rubber sheet but they absorb the time dimension (look at singularity and time dimensions in light cone bending toward singularity). I hope you understand my explains. 43