You can find the requirements over here: https://summoned-by-accident.fandom.com/wiki/Player%27s_House#Quests
In short,the requirements are you'll have to have met Kuri in the Afternoon in the Park. See the Tanuki In The Park (Kuri Intro Quest) and its requirements here: https://summoned-by-accident.fandom.com/wiki/Park#Quests
And then you'll also need to be level 2, which is done by fighting the Drunk Wolf once, leveling up with someone (Zorro, Trald, or Xav), then winning a fight against the Street Horse or Street Rat three times and leveling up again with the same trainer you picked the first time.
Then you'll get a call from Zorro in a random event to tell you to go talk to him. Do so and pick the new talk option to start the quest. You'll also need to have done the first Hangout event with one of the characters in order to take them with you (Ferron's the only one that doesn't count for that).