A set is said to have content iff it's boundary have content zero.
So does compact set have always content?
I can't really find a way to proof this, but for all examples that I can think of
(cantor set, {1,1/2,1/4,1/8...} ) that have weird behaviors and compact, they have content.
The set $[0,1]$ is compact, but has non-zero content. The set $\{0\} \cup \{1,1/2,1/3,\dots\}$, on the other hand, is compact with zero content. You might also want to verify that any finite set is compact with zero content.
The cantor set is an example of a compact set with non-zero content. Note also that the Cantor set is equal to its own boundary.