What longest time ever was achieved at holding light in a closed volume? For what longest possible time it was possible to hold light in a closed volume with mirrored walls? 
I would be most interested for results with empty volume but results with solid-state volume may be also interesting.
 A: The lifetime of a photon in a resonant cavity is pretty trivial to compute, given the cavity length, internal losses, and mirror reflectivity. Switching momentarily to a wave description, we will let $L$ be the cavity length, $R_1$ and $R_2$ be the reflectivity of mirrors 1 and 3 respectively, and $T_i$ be the loss in the cavity medium. Clearly the intensity of a pulse of light in the cavity will exponentially decay, and the lifetime $\tau_c$ (defined by the $1 \over e$ threshold) can trivially be computed to be
$$ \tau_c = -\frac{2 L}{c \ln{\left[R_1 R_2 (1-T_i)^2\right]}} $$
For a one meter cavity with no internal loss and 99% reflective mirrors, this gives a lifetime of roughly 330 ns.
There are much longer cavities, and the reflectivity of dielectric mirrors can have quite a few more "9"s tacked on. For example, the LIGO cavity is something on the order of a kilometer, and if we pretend that the mirrors are 99.999% reflective (that's three"9"s after the decimal place) 1 we get a lifetime of 0.333 seconds (wow).
Lifetime increases rapidly with mirror reflectivity once you get above 99%, so you'll see that if you repeat that calculation with $R=99.9999\%$, you get $\tau_c = 3.333$ seconds. That's an absurdly long time, but of course that fourth "9" after the decimal place is really starting to get unrealistic as well.
1: This is very much an order-of-magnitude guess. I'm not sure of the exact length of the LIGO cavity, and in fact the mirrors are not highly reflective because they are doing a trick called "Power Recycling," which ends up giving them a longer photon lifetime anyway. However, 99.999% is an impressive, although NOT an unrealistic number for a modern high quality dielectric mirror.
A: The photon lifetime in the LIGO interferometers is about 1 s. They use coupled cavities to get the storage time so high; the Michelson arms are composed of Fabry-Perot cavities and an additional mirror at the bright port of the Michelson (the power recycling mirror) forms a coupled cavity with them.  The wavelength of the light is 1064 nm.
All of these answers are only referring to the statistical storage of photons in an optical cavity.  You couldn't, for instance, select a single photon, store it for some time, and then use it later.  This is the type of storage you would need for quantum computing with photons.
A schematic of the Advanced LIGO interferometers http://www.bssaonline.org/content/99/2B/980/F2.large.jpg
A: The set-up you are describing is essentially an optical cavity, and you are asking what is the longest lifetime which has been achieved in such a cavity.
In this paper (also described here), S. Kuhr et. al. describe a supraconducting cavity with a 130 ms lifetime. It is essentially 2 curved mirrors face to face. It works in microwaves (51 GHz), which has a long wavelength (6 mm), and this makes the manufacturing of mirror smooth at this scale much easier. This cavity is one of the key-element of this lab CQED (Cavity Quantum Electrodynamics) experiments. 
I do not know if it is the best one, but I'm pretty sure it's more or less the state of the art in this domain.
A: My answer is not much different from the other ones, but it is based on my personal experience. For several years I have been using Cavity ring-down spectrosopy(CRDS) - the method to measure very weak light absorption that is based on building such optical cavity and placing an absorbing sample inside. The light pulse will pass through the sample thousands of times, thus greatly enhancing its absorption.
For highly reflective mirrors the formula given by Colin K can be approximated as
$$ \tau_0=\frac{L}{c\left ( 1-R \right )} $$
State-of-the art mirrors for CRDS in the visible range can have reflectivity up to 99.999% (link) resulting in a decay time of $167 \mu s$ for $L=0.5m$. During this time the light travels 50000 km which means that it is reflected 100000 times. 
PS: This experiment is done in vacuum (below 1 mbar). At atmospheric pressure the decay time is reduced to a couple of microseconds due to Rayleigh scattering (or absorption by dust particles).
A: This should be a comment but it will be too long.
edit: "I had said :I do not think anybody has designed and done such an experiment, so you could consider it a challenge and do it yourself." I was wrong in that, as an experiment is described by Frederic..
One can make some estimates: reflectivity will play a role. One can make highly reflective mirrors, but still some absorption will take place and the velocity of light is high. Even 10^-5 loss per hit  will rapidly increase absorption in a second. Speed of light is 3*10^10cm/sec so in a 10cm box , thats a lot of hits per second.
In addition each hit by reversing the direction of the photon will transfer some momentum to the atoms/crystal lattice of the surface of the  box and thus there will be a wavelength increase which will finally take it to infrared and invisibility. This needs numbers and calculations, but again, due to the high number of hits per second I expect that the time will be very small by which  the initial photons will be completely absorbed by what is a perfect black box.
