We have the technology to make such a device but you wouldn't be able to achieve such a low cost. We have motors, sensors, and processors that can be built at that scale but it is expensive.
There are two other limitations: energy density and vision. The robots would have to be powered remotely, either by a wire, running current through the surface they operate on, or via wireless power transmission. None are ideal for what you are trying accomplish. The second limitation, vision, demands you have an external vision system, like a webcam, "seeing" for the robots.
It would be a tremendous challenge to engineer such a robot and it would be entirely pointless. The cost of such a device would easily exceed any entertainment value gained from building it so small. You could make a larger version but bringing the cost down to $100 would result in significant feature and/or performance losses. It's much wiser to make purpose-built robots for specific functions, like assembling blocks or interacting with human subjects, without the arbitrary size limitations.