Archaeologists use many different methods to estimate the age of artifacts and/or soil strata that contain artifacts. One of the most important ways is to observe changes in technology: methods used to make tools change with time. There is a big difference between the hand axes made during the Paleolithic and fine arrow points made a few hundred years ago. The technology used to make the Shroud was much different than that used during medieval times or modern times.
Stone tools hydrate and form a patina. Its thickness indicates age. Similarly, all organic materials tend to decompose or change structure with time. Proteins undergo "racemization." Their amino acids change their optical properties. This would apply to the blood on the Shroud.
The DNA in blood and tissue samples degrades with time. The DNA in Shroud blood samples shows the effects of significant aging: only short lengths of the chain remain intact. The reported ABO typing results are very suspect and probably not valid. However, the results prove appreciable age for the Shroud.
Crystalline materials undergo damage that is caused by natural sources of radiation, and Shroud fibers show some evidence for changes in their crystal structure.
Some compounds like lignin change composition with time. The lignin in the Shroud does not give the normal microchemical test for vanillin, indicating that it is quite old. Measurements of the chemical rate for loss of vanillin estimates an age for the Shroud of more than 1300 years, depending on storage conditions.