Upon closer examination, it appears to have a paperboard core with plastic laminated on either side. The surface is badly cracked, and the cracks follow along the grooves in many places.
On the face of it, considering the age and description, and (you might laugh about this) this sounds like one of the supposedly 'unbreakable' records that were actually produced from about 1904 onwards. I've never seen one, but the construction of them appears to be celluloid on a pasteboard base. Vinyl was introduced as a medium in about 1940, so yours would be quite a late example - if this is indeed what it is.
Apparently they suffered from appalling surface noise problems even when they were new, so don't expect miracles.