Letting $\mathbb{Q}$ be equipped with the Euclidean metric.
What I can work out is that it is bounded as its contained in the closed ball of radius ${\sqrt2}/{2}$ centred at ${\sqrt2}/{2} $.
Its not compact as it can be expressed as union of the two disjoint open sets $[0,{\sqrt2}/{2}) $and$ ({\sqrt2}/{2}, \sqrt2)$ (though I'm not sure if this makes it not compact in $\mathbb{Q}$ or just $\mathbb{R}$).
And its not closed as the sequence of truncations of $\sqrt2$ in $\mathbb{Q}$ converges to $\sqrt2$