For natural wood that you want to look like wood, think staining, not painting.
To make that chest look like an old, beaten-up treasure chest, get two colors of stain: one medium-toned one, such as maple or chestnut, and one darker one, like dark walnut or maybe even ebony. For such a small box, you won't need much stain, especially of the darker color, so see if you can get sample sizes. You'll also need some good varnish or polyurethane, not too glossy.
First, stain the whole box with your lighter stain, following the instructions on the package.
Once that's dry, add some realistic-looking damage or weathering: drop the box on its corners a few times to dent/round them off, get something hard and heavy and drop it on top of the box to add dents and gouges, etc.: get creative, but not too creative.
Now, use the dark stain to add "dirt": rub the stain over the box, then immediately wipe it off, but don't do too good a job. The dark stain will remain in the dents and gouges, but not on the surface. Leave more dark stain along the edges and around the hardware. Use a toothbrush to flick stain onto the box to imitate mildew spots. Look up some pictures of the type of old chest you're trying to imitate to figure out where the "dirt" should go.
Once the darker stain has dried as well, and you're reasonably satisfied with how it looks, finish the box with a varnish or polyurethane coating. This is what will make it durable; it will also deepen the colors of the stains. If the result starts looking unrealistic, wait for the varnish to dry, then basically sand it off (or mostly off), adjust the stain, let dry, and varnish again.
If you want to reproduce the metal strapping of the classical "treasure chest", you have several options.
Option 1, you can get what's called "tooling foil" or "embossing foil" at the craft store. It's basically like aluminum foil, only much heavier than even the heavy-duty foil you can buy for culinary purposes. However, it's still light enough that it can be pretty easily cut with ordinary scissors (well, OK, so don't use your nice Gingher fabric scissors - steal the kids' craft scissors instead). To use it, cut out strips in the shape you want, and glue or nail them to your box. You can then use your wood stain to give the metal a more distressed finish. If the stain doesn't seem to be sticking, rub the metal with steel wool to remove the protective finish, and then try again.
Option 2, you can try doing the same thing as option 1, just using ordinary heavy-duty aluminum foil and glue. You can even spray-paint the foil with brass-colored paint. The advantage of this, besides being cheaper, is that the thickness of regular aluminum foil is more in proportion with your small box than the 36 gauge stuff, so you can go ahead and, say, wrap the edges of the opening in a realistic-looking manner, without danger of preventing the box from closing properly.
Option 3, you can get some brass-colored craft paint and paint on the strapping. Do this after staining but before varnishing, and then varnish everything once both the stain and the paint are dry.