We can store cold (ice), heat (i.e. hot water bag) and electrical charge (batteries). We can even "store" a magnetic field in a magnet. We can convert light into energy and then, if we want, back to light. But we can't store light in form of light in significant amounts. What is the explanation of that in physics terms?
For the photons that make up light to exist they have to be travelling at the speed of light. This means that to store them you have to put them in a container where they can move around at the speed of light until you want to let them out.
You could build the container out of mirrors, but no mirror we can build is 100% reflective, or indeed can be 100% reflective. Usually when a photon "hits" the mirror it is absorbed by one of the atoms in the mirror and then re-emitted back out into the container. However, occasionally the photon either won't get re-emitted (leaving the atom in an excited state) or it doesn't hit one of the atoms and makes it way through the mirror and out of the container.
While the chances of this happening for an individual photon are low, there are lots of photons travelling very fast so it happens many times thus causing the light to "leak" or decay.
Building a near perfect mirror is hard, so it's easier to convert the light into something that can be stored and then convert that back into light when need it.
Your examples are a bit misleading. For example you say:
We can store cold (ice),heat (i.e. hot water bag)
But we can only store heat temporarily, just as we can only store light temporarily. Your ice pack will eventually heat up and your hot water bottle will eventually cool down, just as light stored between two mirrors will eventually escape.
and electrical charge (batteries)
Charge isn't stored as charge in a battery. A chemical reaction generates the charge. This would be the same as converting the light to something else, storing that something else then regenerating the light when needed.
We can even "store" a magnetic field in a magnet.
Not the same thing, as we are not storing magnetic charge in a magnet.
A method of storage that might just fit your criteria is storing light in a Bose Einstein condensate. Light pulses can be brought to a halt in a BEC, and in principle stored indefinitely.
I'll try to answer your question in the spirit of how you asked it. Basically you can't really store anything you've mentioned. The ice will eventually heat up, the heat will cool down and the battery will lose charge. A box of mirrors with light shined in it will "store" light like your other examples but it will lose the energy much faster then any of them. Imagine throwing a superball into a box and closing the lid, does it bounce forever and "store" the energy?
We can store light - just for very small amounts of time. I'm no physicist though, so perhaps this link isn't what you intended?
It's hard to store light as light because the most common way light interacts with matter is through absorption and emission, which is how mirrors work. However light rays can be bent by gravity, so it would be possible to arrange several massive stars in a way such that a light ray would move in a loop around the stars without energy loss.
The answer by John Rennie and subsequent comments reminded me of this TEDtalk about energy storage from light.
I don't know the details, but this is what I understand they did: they've studied the electronic and absorption properties of foils made of nanotubes, in particular when combined with the result of some impressive research on infrared imagery. The combined product gains the extraordinary property that it can absorb light and store the energy for longer periods of time and in a cleaner way than batteries (our main and perhaps only real method for energy storage). This energy could be free (because you could just attach these flexible foils to your window for example) and it could even be shared through the coherent re-emission of light (from your window to your neighbours window for example).
It's not storing light in the form of light - what the question asked for - but I think it's as close as we're able to get to storing sunlight for a semi-long period of time efficiently and conveniently, something photovoltaic cells are still struggling with. I imagine Justin Hall-Tipping is making it sound more advanced than it is at this time, but nonetheless it has some great potential and I think it's definitely useful to mention it here. I reiterate that I'm unaware of the details and am not an expert in this field though.
Isn't that what we call "heat" is just a form of electromagnetic waves in the infrared spectrum? So we may say that we can store an electromagnetic wave in a certain media, like water in a thermos. But the first thing, we do not use a thermos filled with hot water as an energy storing container, we just use it for a very simple purpose, to store hot water! But if we were using it as, let's say, a kind of a battery, with a some kind of devise inside that would convert heat into electricity or a mechanical power, to move a vehicle, that would be a different story. So, may be, the very first thing that we need is to find such a media to store the sun light, as that hot gas containing atoms of rubidium or may be that should be some sort of a solid matter, and a second step is to create a sort of a convertor to transform that collected energy into a mechanical or electrical power. By the way, filling a car tank with hot boiling water instead of gas sounds like not a bad idea!))