If you're limited to historically accurate mediaeval tech level and wood is too expensive for even your Empire to afford, you're not going to be building very many ships. The best option for you, if you insist on having a navy, is to bootstrap a navy through alternative means and get your wood in counter-raids from the pirate lands.
Because you tagged this "medieval", I'll be thinking in terms of galleys and combat based primarily on boarding actions. Actually, it turns out that you'll be wanting to do that anyway, because the best seaworthy ships you're likely to get your hands on are the ones used by the pirates.
What you should probably be thinking about are boats rather than ships. Without significant technological advances, you won't even get close to producing the amount of steel you'd need for steel hulls (not to mention binding it all together), but you can use alternative materials to build some boats that probably won't topple over if the weather is favourable.
Your first option is bamboo, or a similar plant. Though you can technically build a hovel out of it, it's not something you'd want to spend a winter in; on the other hand, larger diameter bamboo trunks will float, so you can use them to make a boat that will accomodate a couple of soldiers. If you have bamboo, it will also make a half-decent mast material (if you cut and bind it right to keep it from flexing too much).
Another option that comes to mind is rubber. Though producing it would require you to have the right kind of trees, you don't need to fell them to make something out of the rubber, and since the original vulcanization process was discovered mostly by trial and error, you don't need to know much chemistry either. Vulcanize some rubber with sulphur and have someone blow into it through a glass tube to get some air bubbles inside; this should get you a batch of rubber foam that should float (for a while). Playing around with molds could again let you make a boat that's good enough for near-shore boarding.
If you have the right kind of volcano on your island, you could find pumice deposits somewhere. Pumice is a highly porous rock that's actually less dense than water, so you could conceivably fashion boats out of it, but they're going to be less than seaworthy. Construction-wise, it's stone, so yes, people could use it in construction, but that goes for any other stone as well, and I'll go ahead and assume your empire is not entirely devoid of that.
Finally, and this starts smelling a bit like fantasy now, you could try developing fiberglass. If you can make glass wool (difficult, but not inconcievable) and if you happen to have some sort of resin that you can either directly use as, or refine into, a coating and binding agent, you can try producing fiberglass that will be brittle, but reasonably watertight and good enough to make small boats out of. Again, this is leaning towards fantasy now and requires some development in "alchemy", but with some author intervention, the ingredients are there.
What do I do with all these bloody boats?
Why, you board and capture attacking pirate ships, of course. Hopefully your soldier's lives are cheaper than wood would be, so you can throw however many you feel like at attacking incoming pirates and just keep their ships for future use (prize warfare - capturing enemy ships for your own use - was a thing well into the 19th century). As long as they keep coming, you're building up a navy (and if they decide to stop coming, you no longer need one anyway).
When your navy is sufficiently built up, you load your ships with troops and lumberjacks, and organize an expedition into the pirate lands. You don't have to stay long, so as not to give the natives too much of an opportunity to attack, just long enough to get a few shiploads of (probably extremely expensive) wood.
You ship this wood home and sell some of it for exorbitant sums to help finance your shipbuilding industry. Congratulations, you have now sucesfully started importing wood, and can continue doing so for as long as you like.
You have a choice of keeping up the raids, or trying to establish more permanent outposts to fell timber. Even given the risks involved, if wood was originally too precious to build ships out of, there will be an extremely powerful economical incentive to establish these; this should help you obtain both financing, and willing mercenaries as guards.
Either way, you should by now have enough of a navy to be able to at least organize convoys to protect the wood shipments, which will be further protected by the fact that their precious cargo is not precious to your pirates at all; after all, why would they risk their lives to steal something that literally grows on trees in their homeland.
EDIT: Well, unfortunately the question changed rather significantly while I was writing my answer (also, it no longer matches its own title). The thing is, any material you can build seagoing ships out of can also be used in housing construction; after all, you can always overturn the hull and live inside. You could get around the problem by using a combination of materials, of which one would be structural and one would float (so that you have no single material that's both load-bearing and insulating); something like the rubber foam or pumice I mentioned above, but like I said, neither of those is something you'd really build a ship out of.