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Storing hard cheese in a refrigerator isn't required in order to preserve it.

Cooling down or even freezing cheese messes up its structure (and therefore its texture as you noted) so it's best to store hard cheese in a dry, cool and dark place like a pantry. If you don't have one, your cellar might suffice as long as the cellar is not humid.

Wrap your cheese into a cheesecloth and you can easily store it as it is in a space meeting aforementioned criteria. If you don't have a cheesecloth (I guess most people don't have one), you can also use a plain white cotton cloth with a moderately sized mesh.

For extended periods, you should soak the cloth with vinegar, for more details see https://goldenagecheese.com/blog/specialty-cheese/storing-cheddar-cheese-without-refrigeration/.

This only works for hard cheese, soft cheese should always be stored in a refrigerator and consumed when it's still fresh.