In phonetics we use below symbols to talk about phones' length. My question is that how do we measure it? In other words, since these terms (long, half-long extra-short) are relative, how do we measure each one and assign itphone to a vowel/consonant?
There is an even richer transcriptional tradition used in Finno-Ugric studies which allows up to 8 length distinctions. You can either get trained in transcribing durational distinctions auditorily, or you can use Praat to determine durations from recordings, assuming you have a theory of landmarks (e.g. what do do with aspiration, how to make decisions about vowel-glide transitions). The method of auditory transcription calls for a comparison set, analogous to Jone performances of cardinal vowels. Owing to interesting phonetic properties of Finno-Ugric languages, there are a number of rather consistent and audible sub-phonemic distinctions, and there is no problem in principle with focusing on a particular duration range which may not be minimally contrastive in the language – but it is still audible. Unfortunately, that seems to be a dying art.
Suppose you had a language with only two reliable and detectable surface durational patterns (whether or not it is phonemic). Using just the above "long / half-long / (regular=short) / extra-short" distinction, people generally assign names depending on their own experiences and their implicit phonological analysis. If the distinction is robust and phonemic, without any other complications, then the difference will probably be called "long / short". If the distinction is non-contrastive and you get only a subset of vowels that are the shortest, they will probably be called "regular" versus "extra-short". There can also be cross-linguistic expectations, where based on other languages you know that short vowels are not usually 50 msc long, so if in the language in question you have a series of vowels that are about 50 msc long, you would be likely to call then "extra-short" (for instance, Chuvash has two extra-short vowels, and it would be odd to say that all vowels in the language besides [e,a] are long). If the longer vowels are not strikingly longer, then you might call them half-long. In other words, the most common terminology is "long" vs. "short", and picking one of the other terms would usually indicate that there is something special about the marked term (either phonetically or phonologically).
The lengths of phones cannot be measured in absolute terms. It is always the relative lengths of phonemes that is decisive. One possibility is comparing the lengths of different vowels in the same phonetic context. Another possibility is comparing the lengths of adjacent sequences of vowels and consonants.
The standard system of transcription used widely by phoneticians and phonologists (IPA) is useful and convenient but not perfect. And it is certainly not universal. How could it be either? The speech signal is continuous, but IPA is an attempt at breaking down that continuous signal into discrete acoustic "events". And speech is gradient along most of the dimensions codified by IPA into categorical contrasts. An example: in English, a small amount of weak aspiration often follows the (phonologically) voiced velar onset stops in stressed syllables, but when that occurs it is usually not transcribed as [kʰ] or [gʰ] but rather just [g]. Phoneticians have agreed that aspiration is worth transcribing in English when it shows up with enough intensity and duration to meet a certain threshold.
Going in the other direction, the same symbol may be used to transcribe many different sounds or acoustic events, as long as a majority of phoneticians agree that those variations are similar enough to be grouped into a single category. The bounds of these categories are different from language to language, so if you are working with a language for the first time, you have to learn where other phoneticians working on that language have agreed those bounds lie. Example: a vowel that you might transcribe as [e] in one language, based on its formant values, may be transcribed as [ɪ] (or even [ɪː]) in the context of a different language.
Because of this lack of universality in the transcription system and the non-absolute nature of acoustic manifestations of phonemic contrasts, knowing when to use one label vs. another is a learned skill, as implied by user6726's suggestion of getting trained in making the distinctions auditorily. But I would argue that even making the distinctions visually by analyzing the spectrogram also requires training. As mach suggests, so many things can effect the absolute duration of a vowel, even within a single language. Below is a none-exhaustive list:
Prosodic factors
Segmental factors
"Paralinguistic" factors (which can affect overall speech rate)
It is practically impossible to make a "comparison set" (again, even for a single language) that controls for all of these factors and can be used with data from different speakers and different recording sessions. But even if you had one, you'd have to know when the tokens you were looking at were "close enough" in duration to the respectively relevant example vowels in the comparison set to be given the same respective labels. And that, in turn, would require training and experience.
Phonetics works with a lot of electronic devices that show graphs of the pronunciation of a word on their screens. By means of such devices the length of a vowel can be seen and measured.
