An oscillating block with air

Relevant wiki: Conservation of Energy

Without any information about energy losses in the spring. The box should oscillate forever. It can be argued that the molecules in the gas, vibrate and and the gas absorbs energy in the form of heat. But how much energy can the gas absorb? If it had been a liquid, would it at some point boil? It feels as the question expected a specific answer, but for a correct universal solution more details are required. Nice problem though

The gas is a red herring. Friction within the spring causes some of the kinetic energy to be dissipated as heat, and the machine runs down. Perpetual motion much?

I think the gas warms up at the extreme ends of the spring as the kinetic energy of the molecules converts into heat according to 1/2·m·v² = 3/2·k·T, where T is increasing due to molecular collisions. The spring does work on the gas to accelerate while the heat gets radiated to surroundings and that is how the energy charge of the initial pull of the spring gets gradually dissipated.

Due to the random motion of air molecules inside the box, the box finally stops if the laboratory is at a gravity - free region.

If the box is in a region with gravitational force then the box will stop more easily.

The way I solved it was by remembering that the gas will accelerate at a slower pace than the solid therefore oscillating at a different frequency, dampening the overall motion. Where the energy goes is likely heat, but knowing that isn't necessary to answer the question in my opinion.

The spring ... due to elastic deformation ... while oscillating is experiencing hysteresis energy losses ... and the air within the cylinder is experiencing minuscule compressions and expansions due to sinusoidal accelerations ... this too produces small amounts of adiabatic heating and resultant kinetic energy losses to the system

Complications: air in box has resonance, and compressibility, and thermal losses. Box has resonance and compressibility and losses. Spring has thermodynamic loses.
If you simplify this by assuming that the spring is lossless, and that the box is implastic, then you still have resonance from the air. This quickly can be simplified into a hybrid 2 degree of freedom spring problem with an additional spring stay that is lateral to the initial motion. The 2 frequency vertical springs will evolve chaotically and disperse momentum into the lateral springs. This represents energy losses, not due to heating, but due to the conversion by angular momentum. Eventually, the chaos will increase to a degree where the momentum is no longer in the primary axis of movement, but distributed evenly across all degrees of freedom. This is equivalent to an ambient heat problem. Entropy wins again, even without Thermo considerations. Please note that this is essentially a macro scale version of how entropy interacts with heat on a molecular scale.

Shouldn't it just keep going towards the spring and stop at the ceiling in a place like space?

The perfect spring and vacuum mean the oscillations continue forever.

BUT - the air in the box has mass and therefore has momentum. When the box goes down the gas compresses at the top and puts drag on the box. The opposite at the bottom.

After a while the oscillations stop (or become negligible, since everything is perfect in the system they never really stop...). The gas has heat that is the energy that was in the spring.

I see the solution simply as the energy of the system goes into heating the gas until it comes to rest. No heat is lost by the spring or the gas as they are both surrounded by a perfect vacuum.

The problem statement isn't quite clear about whether the spring is perfectly elastic with zero losses, but let's assume that for now.

The box contains gas, which is made up of molecules. As the box oscillates, the gas within the box will be mixed. That mixing will burn off kinetic energy causing the box oscillation to eventually slow down and stop.

The gas radiates to the surroundings and absorbs energy from the spring. This creates a sink which will drain the whole system.

The statement said vacuum not taking away gravity. Gravity still exists within a vacuum

It is assumed that the spring has a co-efficient of restitution of 1. So the energy is dissipated in accelerating and braking the gas in the box. Which will heat up and radiate to surroundings. But the potential energy of the spring will reduce proportionally.

Work will be done against the frictional force in the spring along with the elastic force. Work done against the frictional force will be converted to thermal energy and radiated away. As energy is continuously lost the amplitude will gradually teplaced and eventually the system will come to rest.

There's no such thing as perpetual motion so this was a triviality.

When the box moves down the gas moves up inside the box similarly vice versa for upward movement of the box so it keeps on eating up some energy of the box thereby leading to the stoppage of the box even if it oscillates in the vacuum.

An elastic spring is one in which no energy is lost to deformation and thus heat. A perfect vacuum is one in which there are no stray molecules to collide with. However, there is no conceptual difference between the air resistance of a box in an atmosphere and the resistance of an atmosphere within a box. What is air resistance? It is the energy lost to slapping air molecules around. Those molecules hit others, etc, creating heat. Creating heat is converting energy.

In a system which has an ideal spring, a perfect vacuum, a perfectly rigid and reflective box (you all forgot that; the box could flex and create heat as well) filled with an ideal gas, the turbulence caused in the gas turns the kinetic energy of the box and the spring into thermal energy in the gas. So the box slows down, and eventually stops.

There are all sorts of other factors which could make it slow down more quickly, but in the end the gas filling is sufficient to stop it in the end.

I believe that due to the non uniform oscillation, i.e. variation from straight line due to the lack of conservation of angular momentum it will come to a stop. If not gravitational waves may play a part.

The oscillation is essentially creating small pressure waves in the gas, like low amplitude sound waves. The gas would heat maybe a degree or less before the oscillation would stop, depending on the energy of the spring oscillation.

The problem is the response of the fluid to the oscillation, which isn't instantaneous. The response of the force applied on the fluid particles is distributed over time. This causes a dampening of the motion of the system.

The metal spring is not perfectly elastic: repeatedly stretching and relaxing it will cause it to heat, with the heat coming out of the kinetic energy of the system, causing the box to slow and eventually stop regardless of what happens to the gas molecules in the box and regardless of the surrounding vacuum.

Simply think that instead of air there is some other dense fluid.then due to sudden change in direction during oscillation of box will cause that fluid to collide with the wall of the box and it leads to lose of energy and deceleration of the box .Due to air this resistance will be less but then also gradually it will stop the box.

I think that the answer given as correct is a good one, but if you want to be precise, the energy of oscillation will never reach zero. In classical mechanics (plus thermodynamics) energy will be transferred to heat the gas, and the energy of oscillation will decrease indefinitely, but will never reach zero. In quantum mechanics, the energy of oscillation will decrease to the zero point energy, but again the energy of oscillation will not be zero.

I think it will slowly slow down due to friction at the point of contact of the spring and wood, and maybe other places.

My reasoning was that the gas does not respond to the acceleration in phase with the box. This is most obvious at the turning points. When the box is at rest and about to turn the gas inside is still moving towards the upper/lower surface. This means the gas performs some negative work on the box, thus heating up and gradually wasting the mechanical energy of the oscillator.