I didn't find my homebirths to be messy at all.
When the midwife who booked me suggested I have a homebirth (yeah, the idea came from the midwife), my main objection was that it would be too messy (my hospital birth seemed to be really messy, LOL). She assured me that it would be fine.
I decided that I would give birth on top of the bed, with there not being a lot of space in our house at the time. So, I bought a water resistant sheet (papery, non-woven material, but washable), and put that over the made bed. I then put a second set of sheets over the protective one. In addition, the midwives provided lots of inco pads.
When my waters broke at the start of second stage, the midwife was ready with a kidney dish to catch any mess. And any yucky bits all ended up on the inco pads.
After giving birth, I went off to have a shower, and in that time, one of the midwives had stripped the bed of the top two layers and put everything in the washing machine along with whatever I had been wearing. All that was left was a lovely fresh bed, complete with newborn baby.
I think it would be more of a problem if your waters went early, so that you lost fluid with every contraction, and if you felt the need to be ultra mobile and naked from the waist down.
For carpets, one of my midwives said she preferred brown parcel paper, shiny side down. It's very cheap and really does protect, but is very unobtrusive underfoot. The problem with plastic sheeting is that it is very slippery, especially when wet.