Large engineering projects have to deal with the curvature of the earth. Long, straight tunnels used for things like particle accelerators have to be adjusted upward since light beams are nearly perfectly straight, and they would climb the tunnel since a "flat and level" tunnel actually curves "down" to follow the curve of the Earth. Also, large bridge projects are constructed to allow for a slightly larger gap between the tops of support pylons. There's a bridge in the NYC area that measured out as about 1.5 inches wider between the center points of the pylon tops than between the center points of the pylon foundations. Why? Because the curvature of the Earth made the pylon tops diverge.
The thing about watching a sailing ship sink slowly below the horizon is true, for the same reason. But anyone by the ocean can prove the world is spherical just by watching the horizon line where "water meets sky". By carefully viewing that line, you'd notice it would undulate with waves. That's because you'd be seeing the actual wave tops right before the water surface curved below the horizon ... from the curvature of the Earth.