= Solution
Let $C$ be the middle-third <Cantor set> and choose a positive integer $d$ with $d\log2/\log3\ge k$; this is possible for every real $k$, including negative $k$. Set $X=C^d\subset\mathbb R^d$. It is closed and bounded since each factor is compact. If a subset of $X$ is connected, each coordinate projection is a <connected subset> of $C$ and hence a singleton; therefore the subset itself is a singleton. Thus $X$ is <totally disconnected>.
For completeness, the dimension of these <Cartesian powers of the Cantor set> can be checked directly. At stage $n$ there are $2^{dn}$ cubes of side $3^{-n}$, giving vanishing $s$-dimensional cover cost whenever $s>d\log2/\log3$. Conversely put independent equiprobable zero-or-two ternary digits on every coordinate. Each stage-$n$ cube has probability $2^{-dn}$. If $3^{-n}\le r<3^{-n+1}$, a Euclidean ball of radius $r$ meets at most a constant depending on $d$ such cubes. Hence its probability is at most $C_dr^{d\log2/\log3}$. For any sufficiently fine cover, summing this bound over covering sets shows that its cost at exponent $d\log2/\log3$ is bounded below by a positive constant, since total probability is one. These upper and lower <Hausdorff measure> bounds prove
$$
\boxed{\dim_HX=d\frac{\log2}{\log3}\ge k.}
$$
The construction does not claim arbitrarily large <Hausdorff dimension> in one fixed ambient Euclidean dimension.
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