There are some issues with the experimental setup you proposed (apart from the fact that when its temperature is lowered the gas would become a liquid and then a solid - if it's not $^4$He: in that case it will stay a liquid). Let's see why.

In the picture above, I've sketched your experimental setup. The black box must be impermeable to matter in order too keep the perfect vacuum (yes, let's say that we managed to achieve a perfect vacuum in some way) and the red box must be permeable to radiation.
The first problem which arises is this: we want radiation to escape our system, so it must be possible for radiation to be able to pass trough both the red and the black box. But if the environment is at a non-zero temperature, it is going to emit radiation too! In that case our cooling equation will be
$$ \frac{dT}{dt} = -C (T^4-T^4_e)$$
where $C$ is an unimportant constant and $T_e$ is the temperature of the environment. At equilibrium, our system and the environment will reach the same temperature (in that case $dT/dt$ will be $0$).
You could be tempted to make the red box impermeable to radiation, but if you do this there won't be any cooling because the radiation will be trapped inside of your system.
So in order for our system to reach absolute zero, as pointed out by Han-Kwang Nienhuys, you will need the environment to be at absolute zero in the first place. This is troublesome. But let's say that we are going to obtain this in some way, maybe removing the rest of the Universe from existence and leaving only our red box in an infinite perfect vacuum. There would still be two problems.
The first (and again, Han-Kwang Nienhuys pointed this out) is that it would take for your system an infinite time to reach the absolute zero.
The second is more subtle, and it is the following: how are you going to know if the system reached absolute zero? To do so, you are going to need a thermometer that is operational at absolute zero. And even if you have it, it must be able to measure temperature with infinite precision: we want our system to be at $0$ K, not $0 \pm 10^{-80}$ K. Is such a measurement even possible?
So, you see: "reaching absolute zero" is a really troublesome concept. We don't really know wether it is possible or not, but it would rise fundamental theoretical problems. Someone will say: "Wait! But Nernst's postulate says that reaching absolute zero is impossible!".
Well, that's not true. I'll quote Callen (Thermodynamics and an Introduction to Thermostatistics), because he has much more authority than me:
The question of whether the state of precisely zero temperature can be
realized by any process yet undiscovered may well be an unphysical
question, raising profound problems of absolute thermal isolation and
of infinitely precise temperature measurability. The theorem that does
follow from the Nernst postulate is more modest. It states that no
reversible adiabatic process starting at nonzero temperature can
possibly bring a system to zero temperature. This is, in fact, no more
than a simple restatement of the Nernst postulate that the T = 0
isotherm is coincident with the S = 0 adiabat.