The most repeated modern explanation for a hundred and eight is astronomical: that the distance from the earth to the sun is a hundred and eight times the sun's diameter, that the distance to the moon is a hundred and eight times the moon's diameter, and that the ancient traditions encoded this. The claim can be checked, and it should be.
The sun: mean distance 149,597,870 km divided by diameter 1,392,000 km gives 107.5. Because the orbit is elliptical the figure ranges from about 105.7 at perihelion to 109.3 at aphelion. A hundred and eight sits inside that range and is a fair round figure. This part of the claim is essentially true.
The moon: mean distance 384,400 km divided by diameter 3,474.8 km gives 110.6, ranging from about 104.6 at perigee to 116.7 at apogee. The correct round figure is a hundred and eleven, not a hundred and eight, and the mean is off by more than two per cent. This part of the claim is false, and it is repeated as often as the true part.
The historical claim is the weaker one. No Indian astronomical text states either ratio as the reason for the number, and Subhash Kak, who did most to popularise the astronomical framing, concedes that the old Indian manuals do not contain it. The derivations actually offered within the traditions are calendrical and astrological, twenty-seven nakshatras times four quarters, or twelve signs times nine planets, and even those are more often repeated than sourced.
The honest position is that the solar ratio is a genuine near-coincidence, that the lunar one is wrong, and that neither is why anyone counts a hundred and eight beads.