This path does not exist on my Nexus5 Android 4.4.2.
Yes, it does, for your process at runtime.
For example, this sample project downloads a file to Environment.getExternalStoragePublicDirectory(Environment.DIRECTORY_DOWNLOADS). If you log the location at runtime, when running it on a Nexus 5, it is reported as /storage/emulated/0/Download. And the download succeeds.
If you are looking for /storage/emulated/0 via DDMS or adb shell, you will not find it. For those tools, default external storage is /mnt/shell/emulated/0. Hence, the downloaded file from the above sample appears in the /mnt/shell/emulated/0/Download directory.
AFAIK, the difference is tied to providing separate external storage to secondary accounts.