What would the steps be to get this paint effect on an unfinished armoire?

initially I just clicked on the link and immediately I thought, 'milk paint' so you are right. it is a distressed finish and you obtain it by painting a white undercoat, then blue, let it dry completely between coats. finally, sand down the edge areas where you want it to look as fi the blue paint has wore away with time and finish by rubbing a paste stain around the edges (these come in milk also).

of course try it first on a piece of scrap wood.

paint the armoire with the white adhesion primer, let dry, and then paint with desired color, once fully dried use sandpaper/with orbital sander if available, and or a knife to "rough up" the edges and give it the distressed look. use the knife to scrape along the edges (use a light touch so you dont cut into the wood or scrape too deep) once you are done sanding if you choose to you can use a brown glaze to give it the old dirty look, and lastly use a sealer....to seal the finished product....if you dont seal over the glaze it will only rub off over time. This is called distressing

Yep, milk paint. It can be found a lot of places, and I know Woodcraft stores carry it, but I'm sure you can find it somewhere near you.

It's kind of like Dolly Parton said, "it costs a lot of money to look this cheap". I think it's pretty easy to work with and would be a good project.