Local energy markets could play a very important role in future electric power systems, allowing small electricity producers/consumers to trade within a small geographical area. In this context, an important challenge to allow the proper operation of a local energy market is to devise efficient and economic computational platforms in charge of data storage, data communication, and any other relevant computational requirements. Fortunately, the recent developments in blockchain-based mechanisms open up promising approaches that can support the effective management of local energy markets. In fact, blockchain technology allows, in particular, to carry out decentralised payments without involving a third party in charge of keeping track of the trades performed, which can potentially simplify and enhance the management of a local energy market significantly. Motivated by this, this paper will study the use of the blockchainbased Ethereum platform as the computational support for a local energy market. Specifically, this paper addresses the scalability of this platform in terms of number of participants in the local energy market, and in terms of trading frequency. With this purpose, computer simulations are carried out for a market that consists of day-ahead trading and real-time trading. The day-ahead mechanism trades energy quantities in such a way that flexible demand can be shifted to balance inflexible supply, and the real-time mechanism trades energy in such a way that flexible loads cover deviations from the day-ahead settlement. The results illustrate that the Ethereum platform is sufficiently scalable, allowing to manage a local energy market with a large number of participants and a high trading frequency.