What is the Observatory:

It's the place where the Guardians lives,each members have their own biome 

-Mog's biome: it's a circus
-Tive's biome: it's inside a castle
-Nepptencio's biome: it's inside a cloclwork tower
-Megumi's biome: it's a library

Where is the Observatory:

No one really knows where it is but it's speculated that the Observatory might be somewhere between dimension