While university participation rates among Indigenous Australians have been on the rise in recent years, parity targets remain elusive. In this context, surprisingly little attention has been paid to how aspirations for higher education are formed and nurtured. Studies tend to focus on barriers to access, often in ways that position Indigenous Australians as a homogeneous 'Otherised' group. This paper counter-balances hegemonic narratives of Indigeneity by analysing what Indigenous students from middle-class families say about their aspirations for university. Drawing on interview data collected in 2016 in government schools in New South Wales, Australia, it offers fresh understandings of school-to-university transitions, both empirically and conceptually. Using the lens of four--dimensional social space as conceived by Indigenous scholar Maggie Walter, we illustrate how race and class function in powerful but complex ways to shape the aspirations of two young Indigenous women. We argue that if we are to shift longstanding demographic patterns of university participation, then visible stories of Indigenous success, such as those told here, can help to redefine current discourses on Indigenous aspirations for higher education.