Odds of Getting In
With heavy competition from furniture makers overseas (source), entry-level jobs are not as plentiful as they once were. The good news is that there are still lots of side doors that can get you in. If a job as a straight-up furniture maker is not available, look for similar gigs. Think carpentry, cabinet making, or upholstering—anything that allows you to work with wood and power tools.
You can also get valuable hands-on experience as an apprentice or by completing a vocational course. Put that on your résumé, and you should find more of those front doors opening for you.
What about just opening your own shop from the get-go? Being your own boss and making your own stuff sounds pretty good—and you can do it, if you know a little something about business and have the funds to start up a shop. If you don't have the money and education, though, it's probably better to just dip your toe into this, rather than jump in cannonball-style.