I'm not up on the technology behind cryopreservation, but here's something from the viacord.com website:
"After testing, your baby's cord blood unit is cryopreserved in cryoprotected blood bags at -196 degrees Celsius."
Unless your 50-year-old freezer could reach -196 degrees, I'd say the answer to your question was "yes." :-D
That definitely is a bit cooler than my mom's freezer back then, or mine even today.
Liquid N2 at -196.
77K in egghead jargon.