We present the detection and characterisation of the TOI-1438 multi-planet system discovered by the Transiting Exoplanet Survey Satellite (TESS). To confirm the planetary nature of the candidates and determine their masses, we collected a series of followup observations including high-spectral resolution observations with HARPS-N and HIRES over a period of 5 years. Our combined modelling shows that the K0V star hosts two transiting sub-Neptunes with Rb = 3.04 ± 0.19 R⊕, Rc = 2.75 ± 0.14 R⊕, Mb = 9.4 ± 1.8 M⊕, and Mc =10.6 ± 2.1 M⊕. The orbital periods of planets b and c are 5.1 and 9.4 days, respectively, corresponding to instellations of 145 ± 10 F⊕ and 65 ± 4 F⊕. The bulk densities are 1.8 ± 0.5 g cm‑3 and 2.9 ± 0.7 g cm‑3, respectively, suggesting a volatile-rich interior composition. By combining the planet and stellar parameters, we were able to compute a set of planet interior structure models. Planet b presents a high-metallicity envelope that can accommodate up to 2.5% in H/He in mass, while planet c cannot have more than 0.2% as H/He in mass. For any composition of the core considered (Fe-rock or ice-rock), both planets would require a volatile-rich envelope. In addition to the two planets, the radial velocity (RV) data clearly reveal a third signal, likely coming from a non-transiting planet, with an orbital period of 7.6‑2.4+1.6 years and an RV semi-amplitude of 35‑5+3 m s‑1. Our best-fit model finds a minimum mass of 2.1 ± 0.3 MJ and an eccentricity of 0.25‑0.11+0.08. However, several RV activity indicators also show strong signals at similar periods, suggesting this signal might (partly) originate from stellar activity. More data over a longer period of time are needed to conclusively determine the nature of this signal. If it is confirmed as a triple-planet system, TOI-1438 would be one of the few detected systems to date characterised by an architecture with two small, short-period planets and one massive, long-period planet, where the inner and outer systems are separated by an orbital period ratio of the order of a few hundred.