Technically, yes, you could home-build a fuel cell. It's expensive, and not easy. To help you learn the principles involved, here is a simplistic project.
* What is a Fuel Cell?
o A fuel cell is an electro-chemical cell that converts a fuel's chemical energy into electricity. The byproduct of a hydrogen fuel cell is water, so a hydrogen fuel cell is non-polluting.
o All fuel cells contain two electrodes, called the anode and cathode. They also contain an electrolyte. And, they require a supply of fuel and a catalyst. A catalyst is a chemical that oxidizes your fuel. That means it turns fuel into positive and negatively charged ions. [FYI: the first hydrogen refueling station was in Iceland and served buses made by Daimler-Chrysler.]
o In this project the anode and cathode will be made of platinum wire. The electrolyte will be water, (H2O); and a common 9 volt battery will be used to break the water into hydrogen fuel and an oxygen catalyst. Hydrogen and oxygen could also be collected using solar power, or some other method, or the oxygen could come from the air and the hydrogen from a hydrogen fuel station. In a car, the hydrogen is stored in a special tank.
* Supplies Needed for the Project
o You will need platinum (coated) wire; a glass of water; tape; a piece of wood or plastic slightly wider than the mouth of the glass; A 9 volt battery and the clip it would connect to in a typical device; and a volt meter to show the results.
* Making the Fuel Cell
o Cut two (eight-inch) pieces of platinum wire, and wind each piece into a coil, for the electrodes in the fuel cell.
o Strip the insulation off the leads of the battery clip and attach the wires to the ends of the platinum electrodes.
o Tape the electrodes to a piece of wood or plastic, and tape the wood or plastic across the mouth of a water glass, so the electrodes are mostly in the water, but not the battery connector.
o Touch the volt meter to the poles of the battery clip. The meter should read zero volts or nearly zero. When you are done 'fueling' the cell, you will repeat this test to show there is energy in the cell.
o To 'fuel' the cell you will use the 9 volt battery to cause hydrogen bubbles to cling to one electrode, and oxygen bubbles to cling to the other. Touch the 9 volt battery to the battery clip for a short time. This will split the water at the electrodes into hydrogen and oxygen. You will see the bubbles forming while the battery is attached.
o When you remove the battery, the hydrogen and oxygen bubbles will recombine to 'make' water, and they will produce electricity when they do it. If your hydrogen and oxygen came from another process, it would still combine in essentially the same way.
o Touch the poles of the battery clip with the meter again, and you will see that the process of combining hydrogen and oxygen to make water is producing electricity.
* Note: This is not how a useful fuel cell is made - this used electricity to separate the water, and then let it recombine, only as a fast way to show what happens chemically.