To answer the question that has been weighing on everyone's mind... yes! you can paint wood veneer furniture successfully. And I do realize that the only mind this possibility was weighing on... is mine.
But I'll share with you in case you become inspired with your next homely-garage-sale-find.
I overhauled a dresser for Baby Girl that matched the fore-pictured nightstand. Again, I neglected to take pictures of the "before" of the actual dresser.
At any rate, I sanded the entire piece well, taking the shine off of the veneer, but not sanding down to the "stuff" underneath it (which is, most certainly, not wood).
I primed it with a good primer, using 1 2 3 primer. Then painted two coats of the lovely shade previously used for the chicken house. Then I brush-painted a thin coat of off-white paint, letting the colored paint show.
Used a little mod podge (glue) to line the drawers with fabric.
And, of course, replaced the hardware. These were quite inexpensive at Lowe's.
Sorry for the picture quality. Some day I'll take a break from painting and sewing to brush up on my photography skills.