In this paper, we study electron transfer networks. These are generalisations of electron transport chains, and consist of a set of substrates which can exist in reduced and oxidised forms. The reduced forms can transfer electrons to the oxidised forms, and there are some electron inflow and outflow processes. We show that under mild assumptions, such systems can have only very simple behaviour, with a single globally stable equilibrium. To prove this we show that the Jacobian of the system has negative logarithmic norm in an appropriate norm. From this result, uniqueness and global stability of any equilibrium follows. The results extend, with only minor modifications, to binary interconversion networks, where the only allowed reactions are interconversions between substrates, and inflow/outflow processes.