The construction of binary locally repairable codes based on orthogonal Latin square was proposed, to resolve the problems of low code rate and high computational complexity of locally repairable codes with $ \left( {r,t} \right) $-locality. The incidence matrix was derived according to the corresponding relationship between the orthogonal Latin square elements and the positions of the digital matrix, and then the all symbol-locally repairable codes (AS-LRCs) were constructed based on the incidence matrix above. The AS-LRCs boasted asymptotically boundary conditions in terms of code rate and code length, and desired minimum distance. The single-check locally repairable codes with information $ \left( {r,t = 2} \right) $-locality were constructed by utilizing the incidence matrix concatenated with an identity matrix. The minimum distance and the code rate of the single-check locally repairable codes reached the optimal boundary conditions, making them be optimal locally repairable codes. Considering that there are nodes with high failure rate in practical distributed storage systems, the single-check information symbol-locally repairable codes (IS-LRCs) with high availability were further constructed through the orthogonal Latin square complete groups, of which the availability parameter $ t $ can be selected flexibly, and the robustness and the flexibility of distributed storage systems were improved.