  We construct isometric and conformally isometric embeddings of some
gravitational instantons in $\mathbb{R}^8$ and $\mathbb{R}^7$. In particular we
show that the embedding class of the Einstein--Maxwell instanton due to Burns
is equal to $3$. For $\mathbb{CP}^2$, Eguchi--Hanson and anti-self-dual
Taub-NUT we obtain upper and lower bounds on the embedding class.
