This is achieved from collecting all four available pets in Bastion.
If you miss any of these, they can be bought at the Lost and Found. Don't wait too long, though. Especially with the Steam Bull.
Squirt (Ragged Hood) is obtained very early in the game, on the first level you can travel to called "The Workmen Ward"
Steam Bull (Pyth Plushy) is obtained early, on the "Pyth Orchard" level.
*PLEASE NOTE* It is vital that you either find or buy this BEFORE you pass the point where Zulf leaves the Bastion. If you cannot talk to him about it, you will not get this pet and the achievement is void.
Anklegator (Anklegator Egg) is obtained on the level "Roathus Lagoon". It's fairly easy to miss on an off-beaten section of the map.
Pecker (Pecker Nest) is obtained on the level "Colford Cauldron". Easy to miss, but like with everything else you can buy it from the Lost and Found.
You will need to aquire all of these before the "Attacking Ura Event". It is possible to achieve after, however you run the risk that some of these creatures may die during the event. Better safe than sorry.
Questions/Comments/Suggestions would be greatly appreciated!