In 1996, philosopher Helen Nissenbaum issued a clarion call concerning the erosion of accountability in society due to the ubiquitous delegation of consequential functions to computerized systems. Using the conceptual framing of moral blame, Nissenbaum described four types of barriers to accountability that computerization presented: 1) "many hands," the problem of attributing moral responsibility for outcomes caused by many moral actors; 2) "bugs," a way software developers might shrug off responsibility by suggesting software errors are unavoidable; 3) "computer as scapegoat," shifting blame to computer systems as if they were moral actors; and 4) "ownership without liability," a free pass to the tech industry to deny responsibility for the software they produce. We revisit these four barriers in relation to the recent ascendance of data-driven algorithmic systems--technology often folded under the heading of machine learning (ML) or artificial intelligence (AI)--to uncover the new challenges for accountability that these systems present. We then look ahead to how one might construct and justify a moral, relational framework for holding responsible parties accountable, and argue that the FAccT community is uniquely well-positioned to develop such a framework to weaken the four barriers.