Project construction and experimentation is at the very heart of ham radio. As I said on the home page, I started out as a young lad taking apart old television and radio sets when active components glowed and before components could only be seen under a microscope. I collected components and tubes from these devices to build my own equipment. While today collecting components is nearly impossible becuase of their small size and level of integration, opportunities still exist for the inquisitive ham. Both the Raspberry Pi computer and Arduino microcontroller provide many opportunities for ham radio projects.
Today one of the most common DYI projects is a DSTAR hotspot. Many hams are not within range of a DSTAR repeater or don't want to deal with changing repeaters as they travel mobile. PE1PLM has created a very successful daughter board for the Raspberry Pi. Couple the Raspberry Pi with the Maryland DSTAR software and a cell phone hotspot; and you have very effective DSTAR access independent of any repeater.
Of course, antenna construction remains, and probably always will be, an area for hams to "build their own".