how much water does it take to float something?

If you had a very strong, form-fitting pool, then a 1-mm layer of water would float the ship because the essentially incompressible water would transmit the force to the container.

A thimbleful of water probably would not suffice. Assuming 5 cm³ in a thimbleful, spread out over the whole hull of a battleship, the water layer would be only about 1 atom thick -- very likely not enough to resist the great weight of the ship.

In order for an object to float, the downward force due to gravity, mg, has to be balanced by the buoyancy of the water. The buoyancy of the water is the weight of the water displaced by the volume of the ship that is under water. So you have to have at least the same weight of water as the weight of the ship and then you can talk of having 1mm more than this around the ship.

yes you could float your boat in a thimble of water. It is the water that is displaced by the presence of the boat, not the quantity of water around the boat that will make it float. If you mark the height of the water with the boat floating then remove the floating boat from your snug pool, it will take water equaling the weight of your boat to refill your pool.

The weight of the water displaced by the ship must be greater than the weight of the ship itself. Then there has to be enough water available to match that weight.