There's a slot for characters to equip pets with in the game

Pet General: As aforementioned, pets are not solely aesthetic, and have the ability to boost its owner's sub-stats when equipped. Raising a pet can be both fun and fulfilling for players; something to get your Shaiya money mind off the grind, so to speak.

There are a total of 5 Degrees, or forms, in terms of raising a pet. Pets are sort of like equipment. There's a slot for characters to equip pets with. Just double click your pet to equip it.

When you want to unequip your pet, open your Status window and drag your pet from it to your inventory.

Some pets require that you have a certain amount of reputation before evolving/equipping. To prevent your pet from dying, you may unequip it and store it in your inventory.

However, if you do that, and then re-equip the pet, then the exp the pet was gaining will go back to zero.

So if you have a Madam Rose that's +3 with 50% exp, if you unequip Madam Rose and then re-equip her, she will become +3 with no exp. When you're in town, your pet will neither become hungry nor gain exp. It will also not die.