Yes, you can build a cage for your hammie. There are a few different options depending on how much you truly want to "build" it.
An aquarium with a wire mesh lid makes a suitable home.
Or you can convert a large Rubbermaid-type storage tub. You can make a top for it using wire shelving pieces (you need to be absolutely certain there isn't space for the hamster to escape though).
Or, if you have a small Habitrail-type cage, you can combine it with a storage tub to make an expanded cage/play area for your hamster.
The last option is what I did for our hamster. I found a Safari cage on sale for $7 but since it was way too small, I used the tub to give her extra room.