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The fat Cantor set is a nowhere dense subset of $\mathbb{R}$ with positive Lebesgue measure. My question is, does there exist a $G_\delta$ set dense in the fat Cantor set with Lebesgue measure $0$?
If such a set does exist, is it possible to produce an actual example of it?
Your fat Cantor set is closed, so it's a $G_\delta$ set. If $C$ is any $G_\delta$ subset of $\mathbb R$, there is a $G_\delta$ subset of $C$ which is dense in $C$ and has measure zero. Namely, let $D$ be a countable dense subset of $C$, and let $A$ be a $G_\delta$ set of measure zero containing $D$. Then $A\cap C$ is a $G_\delta$ set of measure zero and is dense in $C$.
To give an explicit example, you would start by defining an explicit fat Cantor set. Next, you need a countable dense subset $D$; you can do that by taking, for each interval $[a,b]$ with rational endpoints such that $[a,b]\cap C\ne\emptyset$, the least element of $[a,b]\cap C$. Next, you need to specify an enumeration of $D$; that is easily obtained an enumeration of the rational intervals $[a,b]$. So now we have $D=\{d_n:n\in\mathbb N\}$, a countable dense subset of $C$. Finally, define $$A=\bigcap_{k=1}^\infty\bigcup_{n=1}^\infty\left(d_n-2^{-n-k},d_n+2^{-n-k}\right).$$
