 Solution

Yes, for a nonnegative integer $g$: the closed orientable surface of genus $g$ has precisely these groups. A standard polygon cell structure has one vertex, $2g$ one-cells and one two-cell attached by a product of commutators. Its integral cellular boundary maps are zero, because each generator occurs once in each orientation in the attaching word. Thus $H_0=H_2=\mathbb{Z}$ and $H_1={\mathbb{Z}}^{2g}$. At genus zero use the sphere.