How does one define double-lines for edge and node shapes in graphviz dot?

Doubled shapes can be done by using [peripheries=2] on the node

Doubled edges can be done by specifying more than one colour for the edge, separated by a colon. In this case use the same colour twice: [color="black:black"] (or, to separate them slightly more, do [color="black:invis:black"])

I got there eventually! Sorry for the “evolutionary” nature of this answer 🙂

So for example

graph G {
    Foo [peripheries=2]
    Foo -- Bar [color="black:white:black"]
}