The electronic properties of the organic superconductor (BEDO-TTF)2 ReO4·(H2O) were investigated by temperature dependent resistivity, ESR, Hall effect and magnetoresistance measurements. Shubnikov-de Haas (SdH) oscillations were observed in magnetic fields up to 24 T in the temperature range 0.5 K to 4.2 K. The electronic band structure of (BEDO-TTF)2 ReO4·(H2O) was calculated by employing the extended Hückel tight binding method on the basis of its room temperature crystal structure. The two observed SdH frequencies of 75 T and 37 T correspond very well with two cross-sectional areas of the hole and electron Fermi surface pockets obtained from the tight binding calculation. From the temperature dependence of the SdH oscillation amplitudes, the cyclotron effective mass (mc) belonging to the larger and smaller pockets were found to be 0.9 m0 and mc=1.15 m0 respectively. Measurements of the angular dependence of the SdH frequencies show no deviation from that expected for a cylindrical Fermi surface. In terms of our tight binding calculations and experimental measurements, probable causes for the 213 K and ∼35 K phase transitions are discussed. The calculations show that (BEDO-TTF)2 ReO4·(H2O) is a two dimensional semimetal but possesses a hidden nesting. The latter is likely to cause an SDW instability leading to the ∼35 K transition. The resistivity drop associated with the 213 K transition is likely to be induced by an abrupt increase in the relaxation time. The excellent agreement between the calculated and experimentally observed Fermi surface implies that, with decreasing temperature below 35 K, (BEDO-TTF)2 ReO4·(H2O) gradually gets out of the SDW state and re-enters the "original" metallic state, in which it becomes superconducting below 2.4 K.