High-density, high-permanence forms of carbon storage are in demand to save storage space on land or at sea while allowing the world to reach its climate targets. Biochar and calcium carbonate are two such forms that have been considered largely separately in the literature for carbon storage. In this paper, we consider how biochar and calcium carbonate might interact when they are used together with cement as part of a carbon storage system, ideally to form a carbon-neutral concrete. The carbon storage system stores atmospherically absorbed CO2 within concrete, thereby reducing carbon in the atmosphere. In addition, such a system will help in reducing cement usage, thus reducing the need for clinker in cement manufacturing and directly reducing CO2 emissions that result from limestone calcination during clinker manufacturing. Another benefit of such a composite storage system is its use in building structures, a use that has positive environmental and social impact. Thus, further research on the properties of this composite material is warranted. This paper explores the literature on the use of biochar combined with calcium carbonate and cement as carbon storage material. The use of recycled carbon aggregates (RCAs) and LC3 concrete as part of this approach is reviewed. The paper also addresses the possible compressive strength range of the biochar–cement–calcium carbonate composite material, along with other performance expectations. Obstacles to scaling the use of carbon-neutral concrete are identified and an array of research directions are presented, with the goal of improving carbon-neutral concrete and its use.