Huge apologies to one and all, in my first version of this answer I mistyped 184 as 84, and then proceeded to do all my calculations based on that. This is the corrected version.
Also, huge thanks for all the up-votes and comments. I'm so glad it's not just me that is enjoying this question immensely.
The radius of the ball in the image is 21 pixels. The length of the fuselage is 184 pixels, which is 50' 7.5" (15.43m) according to widely available statistics. Hence, a pixel represents approximately 84mm
So, the radius of the ball is 21 x 84 = 1764mm = 1.764m
And the volume of the ball (if solid) is 23.0 m³.
For it to be liftable by the helicopter, using Pilothead's quote of 9000lb (4082kg), the density would need to be 177.5 kg/m³. Mild steel, at 7,850kg/m³, is way too heavy. Expanded polystyrene has a density of 28-45kg/m³, so that would be feasible.
However, if the ball were not solid, and were made of mild steel, the total volume of steel permissible would be 0.52m³. At an outer radius of 1.764m the inner radius must be at least 1.750m to keep the volume of metal down to 0.52m³, so the ball would only be 14mm thick (9/16") and would only cost about $3,300.
So, let's try Titanium (darling of the aircraft industry). With a density of 4,506kg/m³, we could have a volume of 0.90m³. That would allow a thickness of 24mm (15/16"). However, the price of that much titanium would be $229,500 (today's prices - 6 years ago, it would have been over a million US dollars).
Aluminium, however, has a density of 2,700kg/m³. So now we could have a volume of 1.51m³, allowing a thickness of 40mm (1 9/16"), and would only cost $9,360
Pumice, at 250kg/m³, would allow a volume of 16.33m³, allowing a thickness of nearly 0.6m (23½"). That would be very sturdy, I reckon. I can't find a price for a solid lump of pumice that size though.
Incidentally, the people in the picture are also about 21 pixels tall, making them around 1.76m (5'9") tall.
This has now taken up about 3 hours of my time, which like all other hours up to now, are hours I'm never going to get back. However, it has been so much fun.
Following a suggestion from Jules in the comments, I've discovered Lithopore 75-150, which is a ridiculously low density foamed concrete. This could give us the magical solid sphere that we want, with sufficient strength to squish multiple personnel without significant damage to itself. At the 75 kg/m³ end of the range, we could possibly even have some spikes, and a non-negligible cable mass too.
More maths and engineering research required.....