Do you mean the height of land above sea level or the depth of the sea bed ? Height above sea level is measured (as Buildersmate says, from a datum in Newlyn), using instruments to measure height differences (known as levels, theodolites, total stations and, nowadays, gps). Depths to the sea bed are measured using echo or sounding equipment (bouncing signals off the sea bed and measuring the time elaped for the sound to return). Prior to this, depths were measured using the lead and line method (a line with evenly spaced knots or bits of different, distinctive, coloured cloth worked into it and attached to a heavy piece of lead, which was tossed overboard and allowed to sink to the sea bed, which gave the sailors the depth).
Sea beds are notoriously difficult to measure accurately because they are often moving around due to wave and current actions, as well as underwater seismic activity.