I was thinking if it is possible to come up with a $\sigma$-finite measure on $\mathbb R$ which is positive on any uncountable set. I think that I have a proof that there is no such measure - but I am not sure if it is formal enough.
The proof: let $\mu$ be such measure, then for any $[a,b]$ such that $a it holds that $\mu([a,b]) = \infty$. To show it, we consider a bijection $f:[a,b]\to K$ where $K = [0,1]\times[0,1]$ and put $ f_x:=f^{-1}([0,1]\times\{x\})\subset[a,b] $ for all $x\in [0,1]$. Clearly, $f_x$ is uncountable and hence $\mu(f_x)>0$. Now, if there is only finitely many $x$ such that $\mu(f_x)>1/n$ for all $n\in\mathbb N$ then we obtain that there are only countably many $x$. As a result, for some $n\in\mathbb N$ there are infinitely many $x$ such that $\mu(f_x)>1/n$ and hence $\mu([a,b])=\infty$.