There is an important difference in intention between decorating and protecting wood. In these terms 'decorated' means something with a painted pattern done to make something look pretty for the sake of it, whereas protected' is block colours or coatings intended to 'protect' the wood ; i.e. stop it rotting etc so the colouring is not as important as the job it does.
Paint: decorated wood
The Vikings Society does not permit painting wood simply for decoration - we do not wish to encourage excessive use of designs and colours which may not be authentic.
Most of the very limited evidence for painted wood is on carved items, with the paint being used to pick out and emphasise the carved relief. There is very little evidence to suggest that things were just painted.
It is permissible to paint a carved relief and highlight the carving. In such instances the amount of paint relative to the timber should leave more than 50% of the timber on show, and the painted parts are restricted to specific bits of the item, e.g. carved and then painted figureheads on tent poles. The colours used must be achievable with authentic pigments.
Paint: protected wood
It is probable that all boxes (indeed all woodwork) would be protected from rotting etc in some way, and this probably did include painting them. One option specifically suitable for a sea chest is to have it painted with pitch. Almost all boats were pitched to stop them from soaking up water and/or rotting. The usual option is that the timber was oiled using linseed oil (a by-product from making linen) or (for the well to do) perhaps waxed using beeswax. It all depends on what the box is to be used for. The Mastermyr chest had tools in it when it was found, but it is thought that it was originally a wardrobe and used to store clothing. You would not want to pitch-paint a chest used to store clothing as it would stain the clothes. A chest used for such purposes would more likely have been oiled, then dried out and thereafter kept dry.
Painting in block colours is a period option. A good example of this is the Sea Stallion replica longship where the boat has different colours painted on each plank. This not only protects the wood but provides its own decoration.
If considering such block painting you have to do your research and be very very selective on colours. Its also worth noting that the type of paint also informs the end product, and most period paints are bound using egg white, linseed oil or milk. These give specific tones and finishes. There was nothing high gloss - although egg white can provide a reasonable gloss, the number of eggs needed to paint a longship with egg white based paint would be prohibitive. Most colours would be earth based; some would rub off as they are not fully chemically bound to the workpiece (like whitewash).
We know that whitewash was made using crushed and burnt seashells, and that black (or near black) was made using soot. Yellows, browns, oranges and even reds can be made in various shades using ochre and/or burnt ochre. Blue can be made from lapis lazuli but it was very expensive and therefore probably very rare. There are other colours made by mixing their available prime colours like we do today.
One specific modern option missing from the authentic palette is varnish. We have no evidence of clear coatings from our period, at least not on timber. Plain egg white was used as a varnish on some manuscripts, but that is specifically different to large scale timber work. They did have turpentine, or a similar product to turpentine that was steamed out of pine logs in a process similar to making charcoal. However, turpentine does not act like a varnish, since it soaks into the timber. It is likely that turpentine would have been used as a paint binder as we know that it was used as an ingredient of pitch.
One chest from the Oseberg ship-burial is decorated with hundreds of silver tinned iron rivets fixed to the chest forming a geometric diamond pattern. Whether this was done to make the box look good or reinforce it, or a mixture of both, we do not know. It is however a classic example of a decorated timber chest/box - i.e. they were usually decorated using something other than paint. This chest is in the University Museum in Oslo.
While one or two of the bowls from York have things burnt into them, these are on the bases and are either the maker's or owner's marks. Such names or trademarks, probably applied using a brand or heated knife, would be a quick and easy way of marking up timber-ware.
Modern decorative pyrography is not permitted. This is a modern technique, and not evidenced from the early medieval period.
Credits: with thanks to Steve Lines