It can be done for the very same reason you mention: given a torus, one can find inside it four linked tori of arbitrarily large relative measure. So one can proceed as in the standard construction of a Cantor set of positive Lebesgue measure.
More explicitly, fix a sequence $r_n\in (0,1)$ such that $\prod_{n=1}^\infty r_n=1/2$ (say). Start with a torus $T_0\subset\mathbb{R}^3$ of unit volume. Replace $T_0$ by the union $T_1$ of four linked tori $T_{1,1},\ldots, T_{1,4}$ whose union has measure $r_1$. Then replace each $T_{1,i}$ by four linked tori $T_{1,i,j}$ contained in $T_{1,i}$ whose union has measure $r_1 r_2$; let $T_2$ be the union of these 16 tori. If you continue inductively, the intersection $\cap_n T_n$ is an Antoine necklace of measure $1/2$.
One needs to be a little careful to ensure that the diameters of the tori that make up $T_n$ go to zero as $n\to\infty$ (which is needed to ensure the resulting set is totally disconnected)