If our universe has extra dimensions, they exist at each point of space, even in the vacuum. These extra dimensions have a certain universal shape and size. The size is almost certainly a tiny one – most like comparable to the Planck length $10^{-35}$ meters but possible to be up to 0.1 microns in some unlikely "large extra dimensions" models.
The shape of the extra dimensions isn't a hypercube. Instead, it is a smooth closed manifold, something like a sphere (of any dimensionality) except that the sphere isn't among the most likely shapes. At most, the shape (manifold) may have some boundaries.
In the realistic models of Nature with extra dimensions, the dimensions are so small that it's impossible to fit a macroscopic piece of matter inside them. They are almost certainly smaller than an atom – even though, as I said, they may hypothetically be as large as a small insect. But in the latter case, the "large extra dimensions", no matter except for the gravitational waves can exist inside these extra dimensions, so nothing may be sensitive.
Because the dimensions are so small, whatever the exact size is, quantum mechanics is important. Quantum mechanics implies that the momentum $p = mv$ of a particle moving in the direction of these extra dimensions isn't a continuous number. Instead, it is an integer multiple of $\hbar/R$ where $\hbar$ is the reduced Planck constant and $R$ is the radius of the extra dimension (assuming it is a circle for simplicity, but the formula is roughly correct for other shapes, too).
A particle moving with 1 unit of this momentum will look just like "another particle species". For example, an electron moving in those extra dimensions – if it can move there at all – would look like a heavier cousin of the electron, basically like a muon. There is some sense in which the electron, muon, and tau (and similarly for other particles organized in generations) are the same particle that just moves differently in the extra dimensions (this is the case for heterotic string theory on Calabi-Yau manifolds, a very sensible detailed model to explain Nature).
For each specific model, there is some size and shape and one may derive what the particles moving in the extra dimensions look like, what their masses are, what they interact with, and how they may be detected. But basically whatever the physics is, it is always possible to decompose the theory in the higher-dimensional space into a "Fourier expansion" and then it becomes a theory governing fields in 4 dimensions. Qualitatively, the theory will look ordinary, 3+1-dimensional to a non-expert, anyway.
The extra dimensions would guarantee that in principle, the number of particle species in 3+1 dimensions has to be infinite and their density as a function of the mass obeys certain power laws.
Discussions about large metallic 4-dimensional cubes and "how they feel" don't seem to belong to physics because there doesn't seem to be any theory compatible with the known experiments that admits macroscopically large metallic 4-dimensional objects.
But if one happened to construct a theory that is viable and allows these objects, these hypercubes etc. would almost certainly be constructed from "different particles" than those we know. All of them would still interact gravitationally with us – higher-dimensional metallic hypercubes etc. could very well manifest themselves as dark matter – but aside from gravity, the interactions would be rather weak for most of the new particle species.
But gravity could be enough to communicate with this new kind of matter if this new kind of matter happened to admit intelligent life. In principle, they could send us signals via gravitational waves and tell us how it feels to be an insect-like higher-dimensional object. Needless to say, this is not the kind of stuff that professional physicists study because it seems unlikely that any models like that exist and even if they do, the opinions about the life in those Universes are extremely speculative.
