Gas giant planets with periods $20~\lt ~P~\lt ~300~\rm d$ orbiting Sun-like stars are a relatively uncommon outcome of planetary formation, and key questions about the nature and formation of this subpopulation remain unanswered. Theoretical models for the location of their formation (in situ or ex situ) and for their subsequent migration predict different outcomes in terms of planet masses and eccentricities, indicating that observations have a key role to play in disentangling their histories. In this work, we present the discovery and confirmation of a pair of long-period Jupiter-sized planets transiting an F7 star: TOI-791 b is a $0.993\pm 0.033\rm ~R_{Jup}$ planet on a $139.29931_{-0.00012}^{+0.00011}~\rm d$ orbit, and TOI-791 c, a $1.155\pm 0.040\rm ~R_{Jup}$ planet on a $232.01570_{-0.00071}^{+0.00067}~\rm d$ orbit. The two planets are within 0.07 per cent of a second-order 5:3 period commensurability leading to transit timing variations (TTVs) of up to 50 min. We confirm their planetary nature using ground-based photometry, including multiple full detections of the $\gt 11~\rm h$ transits of both TOI-791 b and c from Antarctica with ASTEP, making these the longest-duration transits ever observed in their entirety from the ground. Our detailed analysis of the TTV signal allows us to measure dynamical masses for both planets, which yield densities of $\rho _{\rm b}=0.038\pm 0.008 \rm ~g~cm^{-3}$ and $\rho _{\rm c}=0.047\pm 0.006 \rm ~g~cm^{-3}$, indicating that TOI-791 b and c are two of the lowest density giant planets ever detected. While these measurements are robust, further follow-up is needed to fully characterize the TTV signal and the architecture of the system.