Remove any hardware from the closet and take it apart as much as possible. For example, take off the doors or pull out the drawers if it has any. Set all the pieces on a drop cloth to protect the flooring underneath.
Sand the laminate surfaces with 220-grit sandpaper on a palm sander. Sand any small or hard-to-reach areas by hand. Continue sanding until the laminate isn't shiny anymore.
Wipe the laminate with tack cloth to remove the sanding dust. You can also vacuum the surface if you prefer.
Apply a primer to the laminate using a foam paint roller or bristle paintbrush. Closets tend to have a lot of flat surfaces so a foam paint roller will make the job go by faster. Use a primer designed for laminate. Some of these primers will state they are for glossy surfaces. Cover the entire closet surface with the primer and let it cure according to the manufacturer's instructions. This can take up to a week but it is critical to let the primer adhere properly or the paint won't stick well to the surface.
Paint the laminate in the same way as you applied the primer, using a paint that complements the type of primer you used, such as latex paint with latex primer. Allow it to cure according to the package instructions and then apply a second coat.
Spray or brush a polyurethane sealer onto the painted surface. This is key to keeping the paint on for as long as possible. Allow the sealer to dry according to package instructions and then repeat until you have five coats of sealer on the laminate.