Plastic tarp or shower curtains and some bed mats. It can be messy, but it's usually very isolated. Like it won't just be a huge mess all around the room. It's usually just the immediate spot right under you and anywhere you might walk in the immediate aftermath until/just after the placenta is birthed that is likely to get messy.
We had several shower curtains and honestly that would have been more than enough to protect the carpet...unfortunately, I ended up walking around a lot and pushing and where I ended up was not where the shower curtains were and no one thought to go get them and put them (or anything!) under me. So our floor (gave birth onto the carpet) and bed were completely unprotected. But the mess does clean up and the stains do come out. Obviously, getting something to put under you is a good idea (get several so you can have a spare handy in case you don't birth where you think you will). Also, get a container of salt and some floor cleaner (we used Vanish). You can absorb most of it by pouring salt on the stains right after and then a good scrub with carpet cleaner if you do have any stains. You would never know I had a massive blood stain in the middle of my bedroom! Carpets are completely clean and like new again.
(Also, btw, home birth is awesome!)