A metrizable example can be construct as follows. In the plane consider the subset $$X:=\big\{(x,\tfrac{2k+1}{2^n}):k,n\in\mathbb Z,\;x\in\mathbb R\setminus \tfrac1{2^n}\mathbb Z\big\}.$$
It is clear that $X$ contain (countably many) topological copies of the real line. By a back-and-forth argument (a bit tricky) it can be shown that any non-empty open subset of $X$ is homeomorphic to $X$.