The traditional Midwestern American barn is red, often with white trim. In some areas of the country barns are white, and in Kentucky, barns are often black. The weathered colors bear mute testament to old farmers saving money on paint by buying what was cheap and available. Milk paint was common, and could be cheaply colored with stock blood or iron oxide to produce red. Black barns were painted with lamp black, or treated with diesel to preserve the wood, which turns the wood black. Gray is another predominant color, as many farmers wanted to subdue the presence of the large barn.
In a small yard, you may want your outbuildings to take back seat to the house and landscaping. Colors that work well for this are greens and browns. Planting bushes and vines that will partially hide the barn shed will add to the effect and the building will fade into the background after a few years. You may choose to use stain to blend your shed with its surroundings, as this will mellow nicely and not require repainting.
To tie all the buildings on your property together, you may choose to color them all the same. This can create an effect similar to the old New England farms where the house led to the barn that led to the dairy that led to the chicken coop. Done well, it can be quite effective in creating a harmonious overall scheme. You will want to keep in mind that as the buildings are painted, they will all need to be done about the same time to keep the colors from fading unevenly. Another option would be to paint the barn shed in the same color family as the house to create a complementary look.
Before the 19th century, barns were rarely painted. Their rough sawed siding was allowed to weather to a silvery gray. This can still be done, but to preserve the wood, choosing to oil or use clear stain should be considered. Red cedar shakes are another attractive option for covering a shed without adding artificial color.