To achieve that you will need :
a hand saw (japanese saws are the best in my opinion)
wood white glue
sanding paper or powerfull sanding tools
various pieces of wood
size of the pieces actually depend of the pallet you will have. There's so many differents i can't say you will find the same as mine. But it to show you the principle.
Step 1: The main frame first
You want to make sure your pallet is ready before doing any drawers. So take your pallet, if it's wet, let it dry at home for a days or two in order to dry the wood. This will save you some precious time because if you don't dry it, as you will fill the holes, sand and clean, once dryied, cracks will appear.
So make it smooth, make it soft.
After a first general sanding, blow the dust out of it, and use wood filler. Fill all your holes and cracks, make some reparations if there are missing parts on your pallet, depending on the size. Let dry for a day and sand your pallet one more time.
It should be ready for the next part, drawers !