I went and had another peer at that crane and there is nothing of human scale anywhere near the block. This makes it really difficult to estimate the diameter of the cable - if forced to guess I'd say an inch +/- 0.25 maybe.
OK, 1 inch seems about right. A single strand of 1 inch wire rope is rated at 20 tons, so four strands on an 80 ton snatch block makes sense. That's some expensive wire rope. The winches on the front bumpers of Land Rovers use 3/8 wire rope, to give you some idea of the scale factors involved.