After buying enough things from the house contractor, you will be able to purchase Erebus gates for the underworld for 5 diamonds. These spawn randomly like the chaos portals, indicated by the flaming portal in the floor.
They are locked by heat ratings, with a Tartarus Erebus gate requiring 5 heat, the asphodel one requiring 10, and 15 heat for elysium.
These rooms will require you to fight a few enemies without taking any damage for a reward, but if you take damage you will not receive that rooms rewards. Obviously the Tartarus ones are gonna be easiest just by nature of only being 5 heat. Once you successfully complete one and earn the reward the achievement is yours.