The problem with this question (although your question is still a natural one for those thinking about light to ask) is that it mixes the ideal and the real. You describe an ideal situation with your mirrors, but then ask for what would happen in real life. No actual mirror has reflection coefficient of 1 (which would represent 100% reflectivity) and so any insight you get from using such mirrors in a thought experiment would not be physical.
Now you can consider a mirror with a reflection coefficient arbitrarily close to 1, (although a quick Google search leads me to believe anything above 99% is probably wishful thinking) and is this situation some non-zero number of photons would be absorbed every time the incident light hits the mirror surface. For all practical applications, your light in the setup you describe would still vanish quite quickly. The speed of light is very fast (300,000,000 meters per second) and so if your container is small (with a diameter on the order of magnitude of a meter or so), your light will still have interacted with the mirror walls over 300,000,000 times over the course of a second; likely enough time for all your photons to be absorbed.
But don't give up hope yet! There is one very easy way that many lay people aren't aware of to view light long after the source has extinguished. Just step outside on a dark night. With your eyes or a modest optical aid, you can see stars up to 1,000 light years away or more. When you see these stars, you are seeing the light that left those stars however many light years away. For example, Betelgeuse is a good candidate for supernova in the "near" future (within the next million years). At a distance of roughly 650 light years away, even if Betelgeuse were to supernova tonight, we would still see its light appear as it does right now for the next 650 years!