How to repair this chess set
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Wow, such a beautiful board and pieces
Everything depends on your objective: stabilised and somewhat cleaner would be straight-forward but not without risk. I wouldn't paint the pieces.
Start by stabilising the box, gluing the board to the side of the box. Introduce the glue into the gap with something very thin, it doesn't need a lot but you need to try to cover the whole of the surface to be glued. Make sure you can wipe any excess glue off (PVA glue for example wipes off with a damp cloth, resin glues do not clean up so easily, CA glues set very quickly and don't wipe off well, hide glue would be good but is not extremely common), and that you have the right clamp to hold the board down to the side while the glue sets (make sure you have some pad between the clamp and the wood so you don't leave a mark on it. The clamp will squeeze some more glue out.
Cracks in the boards can only really be addressed by more major surgery, try to avoid that.
Its hard to see how thick the veneer is, sanding is a very bad idea with most 20th century veneers, they are too thin. A very fine (000 or 0000) steel wool with gum turpentine could be used to scrub the surface of the box clean and then be wiped off. Raw linseed oil works nicely and leaves a nice finish but I'd worry about the effect on the colour (turpentine will darken it but will dry out again). The risk is that the oil will darken the board and soak in unevenly (this would not be a concern with the sides of the box). You might use water, but that could affect the wood and the finish more than oil or turpentine would, but in different ways (the veneer glue might be water soluble, the timber might move with the added water). A tree-based turpentine may be the safest thing to start with, but it will not leave the nice finish that oil would.
The loss of colour is difficult to deal with without knowing how it happened. If the squares are different woods then cleaning may well fix the problem (damage to the finish could be hiding the colour). If the squares are dyed differently and the dye has been affected by damp, or they have been painted with different colours, or something, then thats harder without ending up with a really obviously repaired or replaced bit.
A finish for it: a traditional hard cabinetmaker's wax or possibly a pure carnauba car wax, both of which are removable with turpentine, (you may decide you want more a radical fix) applied first with fine steel wool to rub it into the surface and rub it off, and then a cotton cloth to remove the last of it and polish. This should be left set up for a few days. Carnauba is shiny, beeswax is more of a satin finish. Its not a really robust finish, but this is a chess board.
Thank you both for the advices. Greg, the objective is as you wrote it, a stabilized and somewhat cleaner set without any major surgeries.
The loss of color affects mainly one side of the board. If I remember correctly my grandfather stored this board on the top of a cabinet, where the sun could reach it sometimes, so the upper side has lost color over time.
The squares are of two different kind of wood as far as I can tell, but the veneer is really thin, less than a mm. Here is a close up picture of a square with a part of the veneer missing:
if they are just faded, can you live with the fading? you can replace the missing chip in the veneer, but that requires either a small piece of veneer of the right thickness, wood and colour, or filling with something, if you go down the filling path you'll probably end up needing to refinish the board because the filler tends to spread. To start I suggest you leave it as it is and only do things to the board that are reversible (which is why I suggested some kind of traditional wax).
I have cleaned the board and the pieces as Mark suggested. It already looks better 
I decided not to fix the painting on the pieces. Those are scars received in battle after all, and I have never heard of a warrior having a plastic surgery. It would be nice to preserve their current state though. Would it be a good idea to use the wax on the pieces that Greg suggested for finishing the board?
If the squares have faded, I can live with that. I do not want to replace any part of the board. What would be the next steps? Scrubbing the surface with steel wool and turpentine, then waxing? Should I do the same with the sides?
Be gentle with it, if its clean enough for you, stop cleaning. You could wax it, the purpose of that is to seal the surface somewhat (A bit of dirt resistance) and provide a nicer looking and feeling finish.
Greg, you said that linseed oil would darken the wood. What if, after the scrubbing, I would apply linseed oil on the dark squares (but not on the whites), and then do the waxing. Would that improve the contrast of the square colors?
Alright, I will start with a gentle scrubbing with steel wool and turpentine, and see what it does.
What would you use instead?
I said that linseed oil might darken the wood, not that it would. Raw linseed oil will soak in more than it will dry. What a mildly abrasive approach like fine steel wool does it remove loose pieces of finish and very stuck-on stuff and turpentine or oil also help dissolve things like oils and waxes that have been previously applied (wax) or built up (oil). This is why some manufacturers suggest applying wax with steel wool. Don't get carried away and go too far. If you've already cleaned it and got most of the way to where you want to be, then stop. If you wax it, put only a very thin coat on before rubbing it off.
Without knowing either the person or the object its a little difficult, these discussions tend to veer wildly from overly aggressive (truly scary discussions a while ago about straightening warped antique chess pieces by cutting and rejoining them, let alone the continuing idea that you might sand a 19th century box-wood chess piece) to too cautious. You and I have a difference of opinion on how aggressive an approach to suggest but I suspect we would not be that far apart if we were doing the work ourselves. The right approach depends on what the finish is, what the condition is and what end result you want. I cleaned a go stone bowl recently with steel wool and turpentine because I didn't want to risk darkening it and it was covered with wax that had darkened over many years through handling (dirt) and age (there was quite a nice wooden bowl under all that dark brown stuff). A few weeks ago I did some work on a box (not a chess board) with similar problems to this one and for cleaning that I used oil and steel wool, but in that case I had rust stains on varnish as well as a box that was coming apart and I didn't really care whether the wood was darkened. I thought about suggesting cleaning this one with detergent, but I am always nervous about moisture on thin veneer and it looked to me to be a bit too battered and dirty for that. I hope the OP ends up happy with the box, I'm glad he's not going to paint the pieces.
Alright, I have cleaned the board. I did not use turpentine, mainly because I could not get any. But I bought a bottle of "Frosch Wood Cleaner" and I cleaned the board with a towel. I managed to remove a lot of dirt.
The reason of the color loss is apparent now. The finish is missing on the most faded parts. I hope waxing will improve this.
I have glued the crack on the side, now it is drying. I am quite optimistic now!
I'm glad you're getting a good result.
I had a look at the ingredients for that cleaner, I hope you glued the board before cleaning it, there's all kinds of weirdness in there (I stopped at the cosmetic skin moisturiser).
Waxing will make the wood only slightly darker, but it will make it smoother and feel nicer. It won't fix the colour problem. The question about the colour is: is it the wood or something applied to the wood that is causing the colour. If its different kinds of wood then the top layer of the wood has been bleached by light or moisture or something and the wax would have some (not a huge amount) good effect on the colour. If its some stain or varnish (varnish is unlikely) then there is not a lot that you can do. Trying to fix a stain is awkward at best, it risks looking funny. As it is, you have a stabilised, cleaned, functional chess board with significant sentimental value, thats a win/win.
It may be that that cleaner has added some shine to the finish.
DONT DO NOTHING MAN, THATS THE BEAUTY OF A WELL WORN SET, GREG HASNT A CLUE......LEAVE IT BE, JUST DUST IT OFF, ITS BEAUTIFUL AS IT SITS
all capitals and abuse, how sweet.
NO ABUSE IN THERE AT ALL, YOU AVNT A CLUE............................GANDALF HAHAHAHAAHAHAHAHAHAHAHA
Ignore him. I have requested moderation.
A college of mine told me that the original finish on the board is most likely shellac, and that I should fix that before (or instead of) waxing the board. He also said, that instead of shellac I should use some modern alternative which is more durable. Any thoughts on that?
Getting the right shade of brown wax may not be perfectly straight forward. The question is really, how much fixing you do and how obvious a fix can you live with? Fixing the shellac will involve colour matching and matching the finish surface to some degree (pristine and shiny next to worn and a bit patchy). Coloured wax would be somewhat reversible (I've never used it) and traditional waxes can be removed with turpentine. Modern waxes contain all kinds of stuff and are harder to get off, which is why they last longer than traditional waxes. Both of us have been suggesting reversible fixes (apart from the glue) because we are conscious that this should not look too repaired.
As to the moderation, I got some messages from the guy too and blocked him.
Hi everyone,
I have inherited a chess board from my grandfather. It is not an expensive one, but has a huge sentimental value. Sadly it is not in good condition. I am looking for some advices to clean and repair it.