Height is indeed part of the job requirement that needs to be met. Generally, the average height for cabin crew, depending on the airline is somewhere between 159cm (5 feet, 2 inches) and 189cm (6 feet, 2 inches). However a lot of airlines now conduct reach tests, where you are required to reach, with your arm extended up to a certain height. You are permitted to stand on tiptoe to carry out this test.
The reason for the restrictions on height and reach are related to safety
A lot of the aircraft’s safety and emergency equipment is stowed in overhead lockers or to the back of deep stowages & cupboards and this is why you are required to have an extended arm reach, to enable you to get hold of the equipment, should the need arise. Airlines have their own minimum arm reach requirement based on the aircraft you will be trained to work on, but on average they will require that you reach within the region of 208cm (6ft 9 inches) and 214cm (7 feet).
Below is a table listing some of the airline minimum requirement height or reach test requirements.
The above table is correct as of 10/2017