How many times line segments can intersect a Jordan curve?

Most* of Jordan curves have this property. Moreover for most of curves you can not see a point on curve from one side if you can see it from the other side.

To construct such example, start with a smooth closed curve $\gamma_0$, note that you can wave it, so it will remain smooth and any point which is visible from one side on distance 1 is not visible from the other side on distance 1.

Given $\varepsilon_1>0$, you may assume in addition that the new curve $\gamma_1$ is $\varepsilon_1$-close to $\gamma_0$; i.e. $$|\gamma_1(t)-\gamma_0(t)|<\varepsilon_1$$ for any $t$.

Repeat this procedure for distances $\tfrac12$, $\tfrac13$ and so on. At each step, choose $\varepsilon_n$ very small, depending on $\gamma_{n-1}$, then the limit $\gamma_\infty$ is a Jordan curve you want.

$*$ "Most" means a G-delta dense set of Jordan curves.