You don't say "take a time off" it is not correct

My thoughts are exactly the same as yours, Nastya. The first two options are fine, of course, but the prescriptivist in me was 100% sure that the third - "take a time off" - was completely unacceptable. We all know that the noun 'time', when used in this sense, is uncountable and therefore cannot take an indefinite article. This is a basic A2/B1 grammar rule which I would have expected to hold good in this case. Like you, I did a quick Google search, fully expecting the millions of instances of "take a time off" to be non-native errors which I would instantly dismiss. I was all ready to tell you that rules are rules, the internet is full of garbage, and you should never trust what Google searches tell you. However, to my surprise, I found dozens of examples of well-written native-speaker sources containing the phrase "take a time off". Maybe it is OK, then. After all, we can say "take a short time off": 'time' in this sense means 'a period' and is countable. So it doesn't seem unreasonable to omit the adjective and say "take a time off", where 'a time' means a certain stretch of time. Well, there you go. You learn something new every day. One more vote for the descriptivists, I'd say.

Your analysis is pretty well exactly the same as mine. I too would consider "time off" to be uncountable, but google shows a lot of matches. It may be that many of those "a time off "matches are not complete noun phrases, but lead into, for example "a time off request", or are part of a title, like "Once upon a time off Broadway". Even so, there's a lot of what I would regard as "non standard usage." It seems some people now like to "take a time off." But then, of course, we can match many phrases in bad grammar with an exact google search. "Ain't misbehavin'" scores a decent number too ... Sometimes, bad grammar remains as slang or informal usage, though sometimes it becomes mainstream and acceptable. Google's ngram viewer may be more helpful here. This searches a "corpus" of published material in English. It shows comparative results more in line with your (and my) expectations. https://books.google.com/ngrams/graph?year_end=2019&year_start=1800&smoothing=3&content=take+time+off%2C+take+some+time+off%2C+take+a+time+off&corpus=26&direct_url=t1%3B%2Ctake%20time%20off%3B%2Cc0%3B.t1%3B%2Ctake%20some%20time%20off%3B%2Cc0%3B.t1%3B%2Ctake%20a%20time%20off%3B%2Cc0#t1%3B%2Ctake%20time%20off%3B%2Cc0%3B.t1%3B%2Ctake%20some%20time%20off%3B%2Cc0%3B.t1%3B%2Ctake%20a%20time%20off%3B%2Cc0

Only the 3rd variant is correct because you must know the meaning of the article "a", which means "one (countable thing) out of many". You never take "some time off your work" – it is always an exact number of days off, a week off, a month off. You can not take "time off" because time is infinite. You can only take a time off, which is "a day off", "a week off", etc. Remember that "time" is the notion of infinite existence, while "a time" is just a limited interval of "time". You can also say "to go traveling" which is the adverb of purpose. In your versions "take and go" is a parallel construction. Both are OK. But the meaning is different.

Thank you, Jillian!