An analytical model is developed that describes oxygen transport and oxygen consumption for small biological structures without a circulatory system. Oxygen inside the organism is transported by diffusion alone. Oxygen transfer towards the organism is retarded by a thin static fluid film at the surface of the organism. The thickness of this film models the outward water conditions, which may range from completely stagnant water conditions to so-called well-stirred water conditions. Oxygen consumption is concentration-independent above a specified threshold concentration (regulator behaviour) and is proportional to the oxygen concentration below this threshold (conformer behaviour). The model takes into account shape and size of the organism and predicts the transition from (pure) regulator behaviour to (pure) conformer behaviour, as well as the mean oxygen consumption rate. Thereby the model facilitates a proper analysis of the physical constraints set on shape and size of organisms without an active internal oxygen transport mechanism. This analysis is carried out in some detail for six characteristic shapes (infinite sheet, cylinder and beam; finite cylinder, sphere and block). In a well-stirred external medium, a flattened shape appears to be the most favourable for oxygen supply, while a compact shape (cube) is more favourable if the external medium is nearly stagnant. The theoretical framework is applied to oxygen consumption data of eight teleost embryos. This reveals relative insensitivity to external flow conditions in some species (e.g., winter flounder, herring), while others appear to rely on external stirring for a proper oxygen supply (e.g., largemouth bass). Interestingly, largemouth bass is the only species in our analysis that exhibits `fin-fanning'.