Yes you can. Simply make drainage holes in all the draws. Use a well balanced soil. Just make sure before you plant the draws are pulled out enough so when you water one will not be spilling on another creating overwatering.
You could also build a raised frame and use the drawers like window boxes, or use them as water reservoirs for potted plants. They can be used to start seedlings in cooler weather or dry herbs in warmer.
If you are talking about the usual plastic drawer units, I don't think you could keep the drawers in the frame of the unit. Once you fill it with soil, it would probably be too heavy and it might tip over if you pulled the drawers out. However, the drawers themselves would make good planters with holes poked in the bottom or, as Judy Capone Mantell said, use them for water reservoirs for either potted plants or seedlings.
I simply use mine for tools and supplies and my knee pads in the garden. You can paint it to blend in and will need drain holes to let out rain water. I have mine in my vegetable garden so I do not have to run to the shed to get my tools. I have a cover over mine (my husband built), so the rain does not come down directly on it. I can also just take the drawer with the tools I need to where I am working.
you could drill holes in the bottom, bury them shallow, put compost in the corners, and you will always have a place to find worms when you go fishing!
Plastic, unless specifically made for outdoors, will eventually weather rot and just crumble apart. I found out the hard way...LOL
I see metal file cabinets used in the landscape once disguised. Carla is right. Plastic breaks down too quickly especially in the sun.