The Stone Rod is a pre-machined material and crafting ingredient required for many crafting recipes.
You can craft 8 Stone Rods in a Processor from either:
1 Block of Stone, or
1 Block of Limestone, or
1 Block of Bedrock
You can also craft 4 Stone Rods in a Processor from 1 Stone Slab.
Like with everything that can be processed in the Processor there is no need to unlock any crafted recipes first.
Stone Rods are no blocks, so they cannot be put into the quick-bar and also cannot be placed into the world.
Stone Rods are required to craft for example (a complete list will follow at a later time): Plows, Taming Collars, Washers, Cooking Stations, Obsidian Swords, Obsidian Armor, Teleporters, Lamps and Beacons, a bunch of furniture (like Stone Chests) and many building-blocks.