The etymologizing in this discussion area is awesome. This word was used back in ancient and koine Gk for "storeroom" or to refer to something stored or kept unused. Another ancient Gk word for storage space was θησαυρός, which was often used for storing agricultural produce (granary). The distinction between the two seems to have been that the latter was more for storing stuff that was particularly valuable, such as stuff that will become food (granary) and material wealth (treasury). If my old Mandeson dictionary (1961) can be trusted, θησαυρός now in modern Gk means "treasure, wealth, riches," while the verb θησαυρίζω means "to hoard, amass wealth, accumulate," but we should hear from modern Gk speaker to be sure.
Θησαυρός was mostly used in Ancient Greek for the places where precious gifts to gods where stored (as far as I know, of course). In modern Greek, θησαυρός has the meaning you wrote. Your dictionary can be trusted with meanings, they haven't changed since 1961. Maybe it uses some katharevousa endings, though (such as ένωσις, πρότασις etc)
I think "closet" should also be an accepted translation of "αποθήκη", unless I am missing some nuance.
I have been struggling to comprehend the distinction between "ντουλάπα" and "αποθήκη", and the way that Duo uses the word 'closet'. After looking at images served up by searching the two words, it seems that "ντουλάπα" most closely resembles the English "wardrobe" and "αποθήκη", as Duo has it, could be a "storage room", but more often means "warehouse". The English word 'closet' doesn't really fit either of these, but is closer to a storage room than to a wardrobe. From Wikipedia:
"A piece of furniture such as a cabinet or chest of drawers serves the same function of storage, but is not a closet, which is an architectural feature rather than a piece of furniture."
OK, I might be completely wrong here but I'd say maybe αποθήκη is any storage space you can/would walk into, anything else would be εντοιχισμένη ντουλάπα, ντουλάπι, etc.
Of course, the hole in this definition is that, if you wanted to, you could walk into a wardrobe too (and have adventures with fauns, witches, etc.). The point is, you normally wouldn't, so maybe what you call a space could also depend on its internal structure too, e.g. if it's full of shelves and you would only open the door to take/leave things, then it would be ντουλάπι, if you took the shelves out and would walk in to leave something, then it would be αποθήκη. Conclusion, it's a very fine and confusing distinction