That looks like linen. All sheets set we have right now are linen although I don't think they're that heavy. But I do have one summer dress that's made of heavy linen and this sheet looks similar to it.

I spent a lot of money for each sets but so far they're worth it. Only the extra blankets/bed covers are cotton.

We wash them with baking soda for pre wash and white vinegar for rinsing. They immediately became soft. They're warm in winter and cooling in summer.

They do have the natural lived in look but it's better as we honestly don't have time to iron our beddings.