There is a construction mentioned in Ransford (*Potential Theory in the Complex Plane*, pg 143) which supplies an uncountable polar set in $\mathbb{C}$. It is reproduced below.

Letting $\{s_n\}_{n}$ be a sequence of positive numbers, $s_n<1$ and $C(s_1,\dots,s_n)$ be the set obtained by removing the middle of each interval in $C(s_1,\dots,s_{n-1})$ an open subinterval of length proportion $s_n$ of the interval (and so on, starting initially with the interval $[0,1]$), finally set

$$C(s)=\cap_{n\geq 1} C(s_1,\dots,s_n).$$

When $s_n=1-(1/2)^{2^n}$, then $C(s)$ is polar.

As mentioned in the comments, general Cantor sets will not work, and it is an open question to determine the capacity of the standard Cantor set, which is estimated $c(E) \sim 0.220949102189507$.

There is not a set which satisfies the second hypothesis. Any subset of the real line of positive Lebesgue measure has positive capacity and so is not polar (see Theorem 5.3.2 in Ransford).

non-polar, sorry. I suppose your second question made me think that you are asking about non-polar sets, because as explained in Josiah Park's answer, the logarithmic potential of the 1-D Lebesgue measure on $E \subset \mathbb{R} \times \{0\}$ is bounded on $E$, so if $E$ has positive Lebesgue measure, it is non-polar. $\endgroup$ – Mateusz Kwaśnicki Oct 15 '19 at 18:49