We cannot really imagine cuboidal space bending. When you squash a cube, you’re really only distorting its two-dimensional faces (or envisage a stack of two-dimensional squares that are each being distorted). In order to really bend a cube you would have to make part of it invisible, since we cannot see the fourth dimension. You would need to send part of it into an invisible dimension. This is precisely why demonstrations of space-time bending are always made on two-dimensional membranes (reducing the number of dimensions by 1), else we could not visualize it at all.
To make it slightly more complicated, if we take the grid to be the closest representation of space, then in a two-dimensional space, a third line cannot intersect the plane traced out by any two, it can only intersect the lines- a further line can only intersect lines, not planes (in one dimension two lines cannot intersect period). So also in three-dimensional space, a fourth line cannot intersect the cube described by three. A plane can intersect other planes in those dimensions, it cannot intersect the space itself, it can only lie within the space. Should it be able to do so, that’s space-bending.
It’s one thing to hold that space-time as a whole is non-Euclidean in three dimensions, but for this to be true of localised distortions of space time is more difficult to accept, one feels that here a fourth dimension must be called upon, or that the dimension itself be gravity.
This is difficult to conceive, but if we take the lines as “directions along which the dimension can extend”, then there is no fourth direction along which three-dimensional space can extend. But apparently space has a fourth direction into which it can bend. We never “see” space bend, because its not being bent into any of the three dimensions. When a satellite orbits the Earth, its travelling along a gravitational well, yet there is not visible distortion in space, and the same for the Earth’s orbit around the Sun. Just like a two dimensional object can perform no act that can bend a two dimensional plane of which it were part, since all its acts would also be two dimensional in the same plane, so three dimensional objects like us can perform no action that will bend three dimensional space into a fourth dimension, since our acts are also three-dimensional. There is nothing for us to “push” space into, just like the two dimensional creature cannot push into a third dimension. That’s also the reason that when an object falls, it does so in a straight line, not curved. Curvature of space is felt by three dimensional objects as a force travelling in a straight line, as all forces do. They only orbit under special conditions like water in the sink. The Solar System was formed from a rotating cloud of gas and dust which spun around a newly forming star, our Sun, at its center.

The planets all formed from this spinning disk-shaped cloud, and continued this rotating course around the Sun after they were formed. if you pour a small amount of water into the sink it goes straight in, it does not swirl. But for reasons that I cannot fully explain here, which have to do with the manner in which angular momentum stabilizes systems, this “whirlpool” is a preferred configuration, as we can see it is the shape of the vast mount of galaxies in the Universe that are formed around gravitational sinks. The force the planets experience is directed straight at the Sun, not circularly around it. The sink is only the 4-D representation of what is a straight line in 3-D. So all those straight lines are pushed out into the 4th dimension like a sink. But obviously in 3-dimensions its flat as a pancake and straight as an arrow.
So gravity is basically a force felt in a straight line (which is the only way a force can be felt), caused by an extra-dimension that in the case of a spherical object is shaped as a cone. Is this dimension a spatial dimension? We’d like to say “yes” because after all a cone is a shape in space. However the truth is that gravity is the least well explained factor in physics, so we cannot give a certain answer here. We cannot illustrate or conceive of this bending except mathematically, because we only perceive things in three dimensions. Either there are at least four spatial extended dimensions or General Relativity is false.
In any case I have no reasons to doubt that my interpretation of contemporary physics is anything but true (I have no intention of presenting pseudo-science if I am shown to be wrong). But this: Did the Universe have only three spatial dimensions, one would not return upon oneself were one to travel to the end of it. One does not do so through travelling through a sphere, but only along the surface of it. The fabric of our space-time is curved as the surface of a sphere, in the 4th dimension.
Gravity is the equivalent of this spatial curvature. Thus although we cannot perceive it, we experience it, and in this sense I think it is right to state that our bodies perceive the 4th spatial dimension even though our eyes do not. Since that force is felt by all matter including even light, we are able to measure that curvature through the gravitational lensing of light. Gravitational lensing of light is at this point extremely well attested, and was first famously proven experimentally by Sir Arthur Eddington in confirmation of Einstein’s predictions of it from his theory of Relativity. It goes as thus- when light from a star reaches us, if it has travelled past a heavy body en route, then its path is bent and it is seen away from its true position as a result. Light curves around massive objects in the same manner that planets orbit them, as though drawn by an invisible force- the curving of spacetime in another dimension.

Now as it turns out, the equations of relativity do no include an extra spatial dimension, and so it is that the distortions in space occur within the three dimensions themselves. This is best visualized (IMHO) a concentration of space in some areas and rarefaction in others, rather than a bending into another dimension. There is a change in the metrics of space such that they are not uniform throughout. It’s like a cake which is badly mixed, or a 3-D cobweb-grid filling a box that is distorted by the presence of an insect within it. Space is not bent out into another dimension, rathe it just happens to be more space-y in some places moreso than others. So if space is a quantity (which is debatable) then there is more of it in some places than others and that is what the distortion consists in, that would be another manner of thinking about it.

The entire Universe, on the other hand, might or might not have an overall curvature. Our best measurements to this moment do not yield any sign of curvature, however like an ant on the globe, this might purely be because we are measuring too small a part of it. This means, for example that if the Universe is “positively” curved then it actually is like the surface of the Earth and you can walk around in it in circles.