Locally repairable code (LRC) is a class of code aimed at local correction of erasures. This code is applied widely in the distributed storage systems. A code with locality r, requires the access of at most r other codeword symbols to recover a symbol from erasure. This paper studies the existence and construction of binary cyclic LRCs with locality r≤3. Based on the theory of defining set of cyclic codes, the paper describes by adopting the dual of LRCs. After a consideration into the constraints among code parameters, LRCs are constructed and optimized. The existence of binary cyclic codes in arbitrary length with locality 1 is proved and the construction of LRCs meeting the Griesmer bound with locality 1 is offered. Judgment on the existence of binary cyclic codes with locality 2 and 3 is put forward in which LRCs in short length with satisfying parameters and locality 2 and 3 are constructed on the basis of binary cyclic codes in length 7≤n≤99. There is much in these results that researchers further study the relationship among locality and other code parameters as well as the construction of LRCs with satisfying parameters in arbitrary length for reference.