I don't think there is a single dominant answer for the best server / database to use with an Elm front-end, but there are some options that could be mentioned.
NoRedInk has a package that helps Elm work with a Rails backend -- I haven't looked at it closely myself:
There are some Elm packages for working with FireBase -- I haven't tried them either, but they look fascinating.
Gizra is doing some interesting work with Elm and a headless Drupal backend:
One kind-of-logical option is working with a Haskell server -- I suppose the Elm package web-site is one example of that approach:
Another logical option would be working with npm and (say) express as a Javascript server. I can't remember seeing an example of that, though.
Finally, if the server can send and receive JSON, Elm can be made to integrate with it. For instance, I've got an app that is Spring + Postgresql at the backend, and I'm working on porting the front-end to Elm:
I'm sure others can think of other approaches -- I've probably missed some obvious options.