Neutral Hydrogen Intensity Mapping (HI IM) surveys will be a powerful new probe of cosmology. However, strong astrophysical foregrounds contaminate the signal and their coupling with instrumental systematics further increases the data cleaning complexity. In this work, we simulate a realistic single-dish HI IM survey of a $5000$~deg$^2$ patch in the $950 - 1400$ MHz range, with both the MID telescope of the SKA Observatory (SKAO) and MeerKAT, its precursor. We include a state-of-the-art HI simulations and explore different foreground models and instrumental effects such as non-homogeneous thermal noise and beam side-lobes. We perform the first Blind Foreground Subtraction Challenge for HI IM on these synthetic data-cubes, aiming to characterise the performance of available foreground cleaning methods with no prior knowledge of the sky components and noise level. Nine foreground cleaning pipelines joined the Challenge, based on statistical source separation algorithms, blind polynomial fitting, and an astrophysical-informed parametric fit to foregrounds. We devise metrics to compare the pipeline performances quantitatively. In general, they can recover the input maps' 2-point statistics within 20 per cent in the range of scales least affected by the telescope beam. However, spurious artefacts appear in the cleaned maps due to interactions between the foreground structure and the beam side-lobes. We conclude that it is fundamental to develop accurate beam deconvolution algorithms and test data post-processing steps carefully before cleaning. This study was performed as part of SKAO preparatory work by the HI IM Focus Group of the SKA Cosmology Science Working Group.