Secret Bases are a Pokémon Shiny Diamond and Shiny Pearl feature that you can use in the Great Dungeon. While you are there, you will be able to place the numerous statues you find as you dig for fossils in that area. The more statues you find, the more you can pose and eventually you will be able to get pedestals. However, before you can do any of this, we will detail how to create secret bases in Pokémon Brilliant Diamond and Brilliant Pearl.
The only way to create secret bases is to unlock the Digger. You will receive this from the old man from Eevopolis who initially gave you the Explorer Kit. He will assign you several tasks, with the last one leading to the Digger Drill where he asks you to go digging for fossils. After you've done that and come back to him, he'll give you the Digger Drill and give you a short tutorial on them.
All you have to do now is go back to the Grand Underground and get close to any of the walls. You can interact with it and a notification will appear asking if you wish to create a secret base there. If you do, you will make a huge hole, which you can enter. While inside the secret base, you can lay down all the statues you collected from the fossil excavation sites and place them here. Each Pokémon statue increases the speed at which certain Pokémon appear in the underground region at that time.