Intuition about the size of $\aleph_k$ with $k>1$ Assuming CH for simplicity, I know of some more or less intuitive way to think about difference in sizes of $\aleph_0$ and $\aleph_1$. The most straightforward is the distinction of natural/rational numbers and real numbers; and the size of information needed to fully describe each rational number and needed to describe each real number.
Almost straightforward variation of this approach uses finite and infinite binary sequences.
Recently I have learned of an even more enlightening picture by a consequence of Baire Category theorem: a fact that every nonempty complete metric space without isolated points is at least of cardinality $\aleph_1$.
Questions: 

1) Are there any natural examples that provide an intuitive picture of
  the size of $\aleph_k$ for $k\in \mathbb{N} : k>1$ ?
  2) Are there any theorems about metric spaces (like the one I listed) that would let one visualize the spaces of higher cardinalities?

 A: Cardinality, unlike groups, graphs, trees, or vector spaces, have no structure.
It's just a big bag with the label "This bag contains $x$ many elements." it means that one does not have a standard way visualizing it. Furthermore, there are structures (fields, vector spaces, ordered sets, etc) of every given cardinality. So the bag allows you to dress its content with any given structure.
When approaching to visualize infinite objects, if there is no perspective size it is nearly impossible to see it in a way which can tell you much. If you put $\aleph_1$ many points on a line, you will not see them differently than $\aleph_0$ many points, unless you put an $\aleph_0$ line by the $\aleph_1$ one. 
This perspective is even more important once you realize that cardinalities may change via inner models or extension of models, which is very similar to how a pebble and a boulder are nearly invisible from outer space. Using forcing, for example, we can ensure that sets of size $\aleph_1$ in the original model have the same size as sets of size $\aleph_0$ in the new model. This means that we "went further back" and now things of size $\aleph_1$ look so small they are about the same size as $\aleph_0$.
However the above notion is important if you want to consider "absoluteness" of visualization, much like you would like to think that the integers and the natural numbers are somewhat absolute when you imagine them. Once you decide to live in a certain model of set theory, you are saying "I am not moving myself, so the perspective will remain the same".
Personally, I have a way to visualize it. I see large sets, with or without the structures. I find it nearly impossible to describe though, and it is some internal idea which allows me to figure things out (it really does help me more often than you'd think, e.g. to understand exactly what Jensen's Covering Lemma means). Since I work a lot with cardinals which are not $\aleph$-numbers, I usually approach to $\aleph$'s as if they were ordinals.
So I would take the scale that I have chosen ($\aleph_0$, some other countable ordinal, or even much much larger cardinals - perhaps inaccessible or so) and try to stretch lines of the size I want to imagine. If you think about them as stretching into more than one dimension then you can explain how the smaller/larger lines you've taken as a scale won't shrink into nothing-ness.

However the $\aleph$ numbers do have some structure. $\aleph_1$ is exactly the number of countable ordinals. In fact the definition of $\aleph^+_\alpha$ is exactly the cardinality of ways to well order a set of size $\aleph_\alpha$ (up to isomorphism, of course).
Of course this does not apply to limit cardinals, for example $\aleph_\omega$ which is the least $\aleph$ number larger than any $\aleph_n$ for finite $n$. At limit points we simply pick the least possible size, if only to allow some sort of continuity.
The above, regardless to being a natural structure on $\aleph$ numbers, is just one example, and you can easily enough return to the mystery bag with the label "Contain $\aleph_n$ many members!".
