In question, will be whether the territory's government should compensate tenants in the remote community of Santa Teresa for distress caused by lack of reasonable living conditions.
"Landlords should be held accountable for making sure rental homes are habitable," said Joel Dignam, executive director of advocacy group Better Renting.
"This means more than just a few walls and a roof, it means people should be able to live a full and decent life in their homes."
The case comes after multiple bodies, including the NT Supreme Court, found the territory government was obliged to provide tenants in Santa Teresa with housing that was not only safe but also reasonably comfortable.
Residents of the Arrernte community near Alice Springs first brought their case to the NT Civil and Administrative Tribunal in 2018.
Many reported dealing with a lack of electricity, hot water, cooking and functioning toilets for extended periods.
Jasmine Cavenagh had a blocked toilet for eight months, while Enid Young, the lead tenant in the case, had no air conditioning for 17 months and no door for five years.
"Aboriginal families living remotely across the territory have been left to live in dilapidated homes ... for far too long," said Skye Thompson, CEO of Aboriginal Housing NT.
The nearly 70 tenants will be represented by Australian Lawyers for Remote Aboriginal Rights (ALRAR) and not-for-profit litigators Grata Fund.
"Housing is a human right and the deplorable state of housing in remote communities, its impacts on health, education and employment opportunities, should not be tolerated in a country as wealthy as Australia," said Dan Kelly, lawyer for ALRAR.
Territory Families, Housing and Communities said in a statement it was committed to ensuring remote public housing was safe, habitable and appropriate but declined to comment further.
It is expected a decision by the High Court will be reached later this year.
