As pointed out above, there is no doubt that depth of field is based (to a significant degree) on a perceptual judgment.
However, it is possible to define an objective scale. Here is an example of how one might do it.
First, define a normal viewing distance. There are lots of choices. Pick one.
Then define a size for the print. There are lots of choices. Pick one.
Then define an amount of allowed "blurriness." There are a number of choices. Pick one. For example, one might base the choice on what amount of blur is barely perceptible by a person with good vision when viewing and the object at normal viewing distance. This amount of blur would correspond to a certain spot size for a slightly out-of-focus point.
Then, calculate the distance fore can and aft of a well focused object that would produce an out-of-focus point of the diameter determined above. This involves a chain of calculations or measurements. In the best case one could do this by a theoretical calculation, assuming a perfect lens. An imperfect lens will always perform worse than this.
That will give you a depth of field.
Is this the only possible approach to defining depth of field? No!
Another approach would be to base the calculation on the minimum spot size of the lens in question. This spot size is limited by aberrations and diffraction. Then, one would arbitrarily pick a factor, let us say 1.4 or so. (Don't like that factor? Then pick another one.) The depth of field would be determined by determining how far fore and aft of an object would allow an object point to be imaged to a size of 1.4 times the well-focused limit.
Both of these approaches can use objective criteria. The criteria may be arbitrary, but they can be well defined.
The first will produce a result that depends, among other things, on the ratio of the photo size (linear dimension) to the viewing distance. If you decide to change that distance (strictly speaking, that ratio), then the depth of field changes.
The second method is independent of factors such as magnification, viewing distance, photo size, etc. I figure that in most cases the second method will give a shallower depth of field than the first method.
By the way, I am not claiming that these methods are those in common use. I only give them to show that one can define depth of field in more than one way, but that the definition can be based on objective criteria. I suspect that most depth of field calculations are based on something closer to the first method than the second.