Galacticare

The earliest that you can unlock this achievement is chapter 8: One Species' Trash as that's when the Xil become available.
You just need to make sure that you have at least 1 doctor of each species in your hospital at the same time. One of each from the following list:

Dyonai
Human
Kouber Baly
Ohde
Xil
Tenki
Vizarj