Humans have three types of blood cells. Platelets (thrombocytes), white blood cells (leukocytes) and red blood cells (erythrocytes). The platelets are responsible for blood clotting, white blood cells are responsible for the body's immunity while red blood cells carry oxygen round the body.
Red blood cells are produced in the bone marrow. When produced, they have nuclei but when maturing, red blood cells eliminate their nuclei and mitochondria in order to have space to carry more oxygen. This makes them lack DNA because DNA is carried in the nucleus. Red blood cells are also unable to divide due to the absence of their nuclei. They have a life span of about 120 days.
Blood is an excellent source of DNA. However, this DNA is not from red cells, but from other cellular components of blood like white cells.