The 6-torus, obtained by identifying the boundaries of a parallelpiped of $\mathbb{R}^6$ (or, more properly, $\mathbb{C}^3$), is a Calabi-Yau manifold.

You may find this review paper useful, though I wrote it as a graduate student almost 20 years ago and there are probably inaccuracies. In addition, there may have been discoveries in the intervening years that are not reflected there. Caveat lector.

The 6-torus, obtained by identifying the boundaries of a parallelpiped of $\mathbb{R}^6$ (or, more properly, $\mathbb{C}^3$), is a Calabi-Yau manifold. Note that one characteristic of a Calabi-Yau manifold is that it is Ricci-flat, which may impose restrictions on whether you can have metrics of the form $S^n \times T^n$; certainly the "naïve" metric on $S^n$ is not Ricci-flat, but as far as I know it is an open question whether there exists a flat metric on $S^n$ for $n \geq 4$.

You may find this review paper useful, though I wrote it as a graduate student almost 20 years ago and there are probably inaccuracies & infelicities. In addition, there may have been discoveries in the intervening years that are not reflected there. Caveat lector.