How to get size of an element/list in Firebase without get it all?

Firebase doesn't have a count operator, so the only way is to download all children or keep a separate <children>_count property in sync. The latter is not a trivial task (see my answer here for one approach and this example Cloud Function), so most often developers likely end up going with the downloads-too-much-data-but-is-trivial approach:

ref.child("messages").on("value", function(snapshot) {
  console.log("There are "+snapshot.numChildren()+" messages");
})

A more efficient way to count the children would be to fire a REST call with shallow=true parameter, which will give you just the keys. See In Firebase, is there a way to get the number of children of a node without loading all the node data?