~~Red blood cells do not contain any DNA, so all the DNA in your blood is contained in white blood cells~~. DNA on it’s own is a relatively stable molecule owing to the double helix and (in eukaryotes) chromatin structures. A pure, dry DNA sample with no contaminants will survive for quite a while at room temperature and in fact this is how DNA is typically shipped in the mail. However long term DNA preservation is not always trivial because of sensitivity to handling and environmental contaminants.
DNA in buffered solution can survive indefinitely at -20 C or -80 C even in non sterile conditions. At these temperatures, enzymatic activity by DNA degrading proteins is essentially zero. The main issue with frozen DNA is actually the process of thawing it when you want to use it - repeated freeze/thaw cycles can cause excessive shear on the DNA strands that mechanically rips apart the molecule.
As for the case of blood, where the DNA is contained in the nucleus of white blood cells as well as in solution, proper cryo storage will require extra measures to protect the integrity of the cells. This involves chemical preservatives like glycerol and EDTA. These frozen stocks are viable at -80 C for an indefinite amount of time, although you may see a decrease in extraction yield as time goes on. This is attributed to cell lysis, which is extremely slow at low temperatures and in a preserving environment, but nonetheless still occurs at a non zero rate. For short term storage, 4 C is sufficient.
Edit: several people have pointed out that DNA does actually exist in the blood outside of white blood cells
Almost indefinitely. Source: I pull DNA from ages old blood samples regularly.
Scientists have had very good luck pulling DNA from mammoths frozen in the tundra - you can read "Bring back the King" if you have any interest in that sort of stuff.
DNA itself is very hardy. We pulled DNA from a decaying carcass, with just bone left behind. It had been in the elements for 6 months or more, and we were able to identify the carcass to an individual animal based on various DNA markers. That's without even trying, really.
Bacteria, fungi, moss and seeds have been extracted from ice cores, with some organisms still being viable and able to be successfully cultured after as long as 157,000 years buried in the ice in [this study](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4009855/
). Frozen DNA doesn't last forever, but it does last for a very long time.