intuitive description of $\beth_2$ cardinality

Question

I was wondering if there is some intuitive way of describing the size of a set of cardinality $\beth_2$ ($2^c$) ? For the natural and real numbers the I think the intuitive understanding is quite easy, as is implied by the names: $\aleph_0$ is "countable" , i.e. we can imagine it as set of discrete objects that can be labeled and counted, while the continuum $c$ is, well, "continuous". Is there any similar intuitive way of explaining in what way $2^c$ is qualitatively different than $c$ ?