Oops---I found the answer to the first question of mine here on Wikipedia---this is really classic material. I'll leave the question open for a bit, in case someone tells me something interesting for my second question.
(A version of the above was mentioned by Michael at about the same time as my "oops", so I accepted his answer)
Sorry for asking a basic and naive question---if this is textbook material somewhere, please let me know, so that I may close this question.
My question is:
When is a metric space $(X,d)$ isometrically embeddable into some Banach space?
Can one say something "stronger", if we know that the closure of $X$ is actually a convex cone?
(By "stronger" one of the things I mean is whether we can actually obtain an embedding that is easy to compute...)