Long duration energy storage systems with 8-12 hours of capacity are one of the best options to reduce the increasing curtailment of renewable energy, being able to provide intra-day storage. Among those systems, Carnot batteries operating with CO2 can be a promising solution due to the relatively high round trip efficiencies (up to 60%), being site-independent, including the possibility to store and sell both cold and hot thermal power in addition to electricity. In this work, the detailed sizing of the main components of a CO2 Carnot battery is proposed: in particular, an interesting and promising feature of the system is represented by in the adoption of the same heat exchangers during the charging and the discharging phase, while the cold and hot storage systems are inspired by the commercial solution proposed by Echogen Power Systems. Specifically, the hot storage consists of two heat transfer fluid loops: a pressurized water loop and a diathermic oil loop, requiring two different insulated tanks, whereas the cold storage is based on an ice slurry tank. A routine in MATLAB has been developed to properly design the system and optimize its main variables to maximize the round-trip efficiency and minimize the storage costs, but also including the calculation of the levelized cost of storage. The battery is simulated with a cold storage at 0°C and hot storage between 71.6°C and 293.7°C, with the cycle maximum pressure of 250 bar and a round-trip efficiency of 54.6%. The specific capital cost of the system is 2033 €/kW_el,ch, with the largest share being the storage systems. The levelized cost of storage is estimated around 0.1 and 0.3 €/kWh, depending on the electricity selling price, and a relatively large internal temperature difference in the hot storage (20°C) is suggested for these systems.