If you only care about adding numbers, you can e.g. do that using water or marbles. You only need to build an XOR gate, an AND gate and an OR gate.

In case of water, the gate will have two inputs as water streams. They should be aligned so that when the streams hit each other, the water fill flow into a cup in the center of the aperture. When the streams don’t hit each other, the water passes the cup into another cup on the bottom. Carrying the water out at the bottom is the XOR gate, carrying the water out at the center is the AND gate and both cups together are the OR gate.

Then it’s just about setting up the circuit and that would be a full adder without electronics.