Obesity Measurement

The most accurate methods to measure a person's body fat is to use a X-ray test called Dual Energy X-ray Absorptiometry, also known as DEXA, or weighing a person underwater gives pretty accurate values. (This method is not practical and involves troublesome procedures).
A chamber that uses air displacement can be used to measure body volume (but rarely required).
However apart from scientific interest these methods are not routinely used or needed.
There are other methods to estimate body fat. One is to measure the thickness of the layer of fat just under the skin in several parts of the body.
Another involves sending a harmless amount of electricity through a person’s body. Results from these methods, however are always inaccurate and do not provide any reliable information which could be used for therapeutic purposes.
BMI (Body Mass Index) or Weight-for-height charts vary as per individual organizations & institutions and are not capable of distinguishing between excess fat and muscle but are fairly doable in most cases unless the person under consideration is an athlete.