I thought I heard my teacher said: 'Don't construct a sentence without a verb.'
Is there such rule in english grammar?
She looks like 70 years old. Perhaps older.
How about the question above? Is it correct? Is there any book that can be used references for wide range of topics in grammar(including syntax, morphology et cetera)? I have finished oxford grammar in use recently and think it wasn't enough for me.
The usual definition of the word 'sentence' demands that there be a verb*, so the question becomes nonsensical.
However, rephrasing to 'Don't use structures other than sentences,' we must say that this is a very shaky rule-of-thumb rather than a grammatical requirement. In conversation, sentence fragments / sentence substitutes such as Tomorrow, Yes, If you must, On the table are often given as answers and are quite acceptable. Follow-on fragments are also quite acceptable in informal writing: She is 70 years old. Perhaps older.
Look up articles on 'sentence fragments' and 'crots' on the internet for guidance on actually using these structures. Overuse is not advisable, and only normal patterns should be used:
Where is the paper?
Where will you go to?
What are you writing about?
*The fact that 'sentence fragments' are sometimes called 'minor sentences' is, I feel, confusing.
"She looks like 70 years old" does contain a verb, so it is a correct sentence, if you are asking if it is correct based on what your teacher told you. It is, however, not natural English. "She looks (about) 70 years old" is better.
As to whether every sentence must contain a verb: this depends. If you are learning English, then probably best to follow this rule. If you are exercising creative license, however, then no: "Yeah.", and many other examples, can be considered a sentence (http://www.merriam-webster.com/dictionary/sentence?show=0&t=1388188840).
