The space $\mathbb{C}/\mathbb{R}_{>0}$ is obtained by adding a point to $S^1$ in such a way that the added point is closed and the topology is the coarsest possible under these conditions (so that a subset is open iff it is an open subset of $S^1$ or the whole space).

The subspace $S^1=\mathbb{C}^\times/\mathbb{R}_{>0}$ of $\mathbb{C}/\mathbb{R}_{>0}$ is open, and the ambient space is contractible (by passing to the quotient the retracting homotopy $(z,t)\mapsto tz$).

As erz said [in the comments](https://mathoverflow.net/questions/392062/is-s1-an-open-subspace-of-a-contractible-space/392066#comment1000272_392062), this is not possible if you want your contractible space to be Hausdorff, because otherwise $S^1$ would be one of its connected components.