Health care systems face resource scarcity that may jeopardise their financial sustainability as well as the quality of delivered health care. In view of that, the association between technical efficiency, access, and quality of services should be investigated, despite some past attempts that led to mixed, unclear, and perhaps biased results. We use a dataset composed of financial resources, hospital services, appropriateness and timeliness of care, patients’ clinical safety, access to health care services, demographics, and epidemiology variables to study the aforementioned link regarding the Portuguese public hospitals (operating between 2013 and 2016). Quality and access data are aggregated into three main composite indicators, through Grey Relational Analysis (GRA). Bias- and environmentally corrected efficiency scores are estimated via bootstrap-based directional Data Envelopment Analysis. A double bootstrap algorithm is employed, using GRA-based quality indicators as predictors of technical efficiency. Evidence suggests that (1) Portuguese public hospitals exhibit low performance in terms of quality, while the different indicators present considerable correlation among them and with hospital size and patients’ complexity characteristics; (2) patients’ clinical safety, appropriateness and timeliness, as well as access to health care services are consistent and significant predictors of technical efficiency; and (3) the association between efficiency, quality, and access depends on the interaction between appropriateness, timeliness, and access. Therefore, quality and access can be improved with no efficiency sacrifice and vice versa.