The simplest, and possibly only, spring only system that would work like that would be made of nonexisteum as the energy would still have to be dealt with, and the power level exceeds what is seen in the video.
This video TFB provides another firing demo which gives a more realistic impression of the recoil level experienced in practice. The shooters take care to place the butt to their shoulders in a conventional manner, even after having fired a number of shots, and the recoil reaction on this bodies can clearly be seen. While the recoil may be substantially lower, it is nowhere nearly as low as "zero recoil" - and also is much lower than you'd expect from the initially cited video, suggesting that he used very low energy loads.
However, a spring plus 'other things' to spread the energy pulses and absorb as much energy as possible would work - and demonstrably does, as the (US, not Russian) AA12 is an entirely real gun.The original AA-12 was a 1972 US design by Maxwell Atkinson (images). It has since been developed by 'Military Police Systems' into the current version.
When the bolt flies back after firing to cycle another round, around
80% of what would normally be felt as recoil is absorbed by a
proprietary gas system. A recoil spring grabs another 10%, leaving the
final recoil a remarkable 10% of the normal recoil for a 12-gauge
round—so you can point the AA-12 at a target and unload the full
magazine without significant loss of accuracy.
Based on the notes below it seems likely that the "proprietary gas system" not only absorbs and stores the energy initially, but must also dissipate energy - presumably by heat transfer via compression. Power level to be dealt with during during firing at full rate of 300 shells/minute is about 150 Watts, or about 600 Joule for a maximum of under 4 seconds for an 18 round drum magazine. 600 Watt.seconds (600 J) is enough to make the gun 'somewhat warm' but low enough to be dealt with realistically.
If the level to be dealt with by the user is 10% of total, as claimed that's a continuous rate of about 15 Watts. That's a level of power input that people would exert for short periods on many hand operated units that take noticeable user power (many not so well known today) - sirens, power generators, window winders ... . Well below the level needed to crank a car by hand (remember those ? :-) ). 12 gauge load energy can vary significantly. Looking at the effort taken in the video I'd guess his loads were not up to maximum power level for 12 gauge.
Overall the user + gun must absorb energy from the explosive charge as they apply an equal and opposite reactive force to the accelerating projectile. The velocity of the overall mass (user + gun) will be low (fortunately) compared to that of the projectile(s). The timing and shape of the reaction can be altered by springs (gas or mechanical) and moving mass but the "reactive energy" must be absorbed either as heat + user absorbtion.
Watching the video portion in question it appears that the gun barrel dips during the period during and after firing rather than rising as I would have expected. This could be achieved by accelerating an internal mass using pre-stored energy (gas, springs, ...) to counterbalance the impulse peak. The energy still needs to be absorbed but with suitable design could be spread across the available time period linearly or in a manner that best suits the users muscle response.
Energy involved varies with load used and can differ by more than 5:1 - so what you see in the video and what you experience with a maximum load can vary considerably. This reference The popular 12 gauge indicates that the upper limit that most
recreational shooters can tolerate from a 12 gauge shotgun is 21 foot-pounds of recoil energy*. That translates to about 3 kg.m or slightly under 30 Watt seconds.
That's not as arcane a unit as it sounds because - the 300 rounds/minute firing rate = 5 rounds/second or a continuous power level at 30 W.s per shell of 30 x 5 = 150 Watts. Providing a sustained 150 Watts is a substantial workload. Most users on an exercise bike would be unhappy to maintain that level for more than a few minutes.
*I took the recoil energy, projectile mass and muzzle velocity figures they quoted on that page and compared calculated kinetic energy of the projectile with recoil energy. I had expected them to correlate approximately linearly but the claimed recoil energy increases much more rapidly than I'd expect from the calculated projectile energy. I tried various values of x in 0.5 x m x V^x without getting a reasonable fit. Using impulse (delta mV) is worse. Interesting. Maybe a better understanding of acceleration profile and burn dynamics is needed. Or, no doubt easier, just measure it.
World guns.ru More history and tehnical details.