Dense embeddings into Euclidean space

The question is a follow-up on this old post. Fix a positive integer $d$ and consider $mathbb{R}^d$ with its usual Euclidean topology. Given a metric space $(X,delta_X)$, what conditions are needed for $(X,delta_X)$ to admit a continuous (preferably bi-Lipschitz) embedding into a dense subset of $mathbb{R}^d$?