Domestic Medium Hair is a coat description, not a breed. It's used for mixed-ancestry cats whose fur falls between short and long, with a fuller tail, tufts between the toes and a soft undercoat, but no heavy ruff. Colors run the whole range, from tabby and tuxedo to calico and solid black.
Size, personality and health follow the cat, not a breed standard. Most weigh 6 to 14 pounds, are sturdy and often live well into their teens. Expect extra brushing, especially in spring and fall when the coat sheds more. Because they come from the general cat population, nearly any household can find one that fits, and meeting the cat is a better guide than the label.