Can we lock the light (photons) up inside a box? Why / Why not? Imagine a box designed to make vacuum state inside. Is it possible for us to lock the photons up inside something completely enclosed?
 A: The speed of light is about $300,000\,\mathrm{km/s}$. A ray of light, even if enclosed in a large box, will bounce off the walls many, many times per second.
Now even if we use the most reflective material to coat the box' walls, like silver or platinum, each time the ray hits one of the walls a small amount of the light is absorbed by the wall's coating.
And many, many of such reflections with absorption every second means the ray is very quickly fully absorbed.
In short, even in a box with super reflecting inside walls it is always dark (unless there's an interior light source).

- A numerical example -
Take a cubic box with a $1\,\mathrm{m}$ side. The question is:

How high would the reflectivity ($R$) of the inner walls have to be so
that after $1\,\mathrm{s}$ the ray of light has retained $10\,\text{%}$
of its initial intensity ($I_0$)?

Answer:
In $1\,\mathrm{s}$ the ray bounces off the walls $N$ times, where:
$$N\approx 3\times 10^8$$
and:
$$\frac{I(N)}{I_0}=R^N=0.1$$
$$\log_{10}R^N=\log_{10}0.1$$
$$N\log_{10}R=\log_{10}0.1$$
$$\log_{10}R=\frac{\log_{10}0.1}{N}=-3.3333\times 10^{-9}$$
$$R=0.999999993$$
So it's a ridiculously high $R$ value, only to have the ray retain $10\,\text{%}$ of the initial intensity during the first $1\,\mathrm{s}$.
A: I'll expand quantitatively on Gert's answer. I do this using a greatly simplified example. Let's take a square box of exactly one cubic meter. That way it becomes a matter of how many times the light can bounce inside the box. The best reflectivity of a material I could find is dielectric mirrors. Wikipedia claims it can reach a reflectivity of 99.999% (or better) but only for a small range of wavelengths. This doesn't matter to us so let's just pick monochromatic light. Using the 99.999% reflectivity you can calculate that it would take about 690772 bounces before the light reaches 0.1% of the original light strength. This is a fair cut-off to call the light completely dimmed.
The speed of light is about $3\cdot10^8$m/s. So it only takes about 0.002 seconds to reach the number of bounces I mentioned previously. For comparison, when a screen has 60FPS there is only 0.017 seconds between each frame. So for us humans it would just look like the light immediately dimmed.
A: No, it is not possible to do, at least not perfectly. Any system complex enough to keep light in a small region is also complex enough to interact thermally with the photons and slowly leak energy to the surrounding environment.
Light carries momentum. Imagine a light wave travelling towards the wall of your imaginary box. If the light is to be contained in the box the wave must reverse direction when it reaches the boundary, and hence change momentum. Since momentum is conserved, something in the wall must gain some momentum and also energy (since gaining momentum always increases kinetic energy).
Thus, either the wave must slowly lose energy to the walls of the box over time, or doesn't have have momentum (in which case it's not really a light wave but a static EM field).
As the box gains energy it will gain some sort of temperature, at which point it will start to act as a black body and radiate light into the environment and back into the box. Slowly the original light you were trying to trap will be converted to thermal black-body light.
You can slow this down by making the inside walls of the box very efficient reflectors (like a superconductor), but you'll never be able to get around the fact that something inside the wall has to absorb momentum and hence energy.
