Since things like happiness, sadness, and grief are highly subjective, so I don't think there's any way you could measure those variables directly.
You could operationally define those emotions, such as measuring happiness by the number of hours someone spends doing something they enjoy, but you can imagine all of the confounds involved with that.
Or, you could define a general scale (see here for a list of those having to do with pain) and rank how people score on it. Then, you could use principles of non-parametric statistics to compare your rankings.
You could also measure known anatomical correlates (e.g., activity in the amygdala as a basis of fear via fMRI), but so many emotions/feelings/other background processing overlap in certain brain regions that the validity would have to be shown through good controls.