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 !
Step 2: Drawers
To begin, in this table, i've used 2 pieces of walnut on each sides of my drawers. I think it make a nice contrast. If you plan to do the same, it's your very first step for the drawers, as it will set the width of your drawer.
Cut 8 pieces and glue/nailed it as shown on the pictures. Most of my wood pieces for this table are used kinda raw. So i try to adapt it as much as possible to avoid cutting with power tools. It's mostly handmade so that almost anyone can build one.
For the drawers front, i've used the same process as the wooden earring from my other instructable, but you can use a plain board as well, up to you, be creative !
Measure the space avelable for your drawer and cut to dimension. Allow a 1/16 of an inch for each side to let it slide freely later.
So i've prepared my drawers, now lets deal with the drawer frame. At this point you need to decide the look of your table. do you want flush drawers? slightly inside? outside ?
For a flush version, mesure the distance between 2 opposite front drawer, if flush then it's the width of your pallet. Substract 2 inchees, then divide by 2, it will give you the total depth of 1 drawer. Measure the parts needed for your frame, and glue your drawers.
To make sure they are square, measure the diagonal of the drawers, they should be equal. Once dryed, glue and nail your drawer bottom, and you are pretty much done !
Make two sliders per drawers from scrap wood. Attached them as shown on the pictures, nail and glue, et voila !
For the top of the table, i've seen many different pallets, so mine got huge gaps between two slats, i decided to fill it with beech wood slats, leaving a tiny space between them. I like the vintage, boxy look it got.