Autistic people experience more co-occurring health conditions and, on average, die younger than non-autistic people. Despite growing awareness of health inequities, autistic people still report barriers to accessing healthcare. We aimed to explore the experiences of autistic people accessing healthcare, shining a light on the complex interplay of relevant factors and to explain, at least in part, the possible reasons underling health disparities and adverse health outcomes. This is a qualitative study from an autistic research team. Data were collected from 1248 autistic adults as part of a large, mixed-methods, international survey exploring barriers to primary healthcare. This article reports the qualitative findings, following a thematic analysis. Using our exploratory findings, we then constructed a model to explain the reported experiences. Respondents reported a variety of barriers. Here, our article gives voice to their stories, in their own words. Themes included: early barriers; communication mismatch; doubt -- in oneself and from doctors; helplessness and fear; and healthcare avoidance and serious adverse health outcomes. Our constructed model outlines a chronological journey through which healthcare access barriers may lead to adverse health outcomes. Our findings also build on the double empathy problem, situating this in a medical context, proposing a triple empathy problem.