In the distributed storage system, locally repairable code can improve repair efficiency when a node is at fault. Quaternary distance optimal codes are easy to realize and their errorcorrecting ability is superior to that of the binary distance optimal codes when code length and dimension are given. However, there are lots of work needed to construct quaternary locally repairable codes by quaternary distance optimal codes. When the dimensions of quaternary distance optimal codes are 2≤k≤4, by combination with the operation of combination, such as extension, the deletion and the juxtaposition, the paper can obtain generator matrices of quaternary Simplex code, MacDonald code and few distance optimal linear codes to construct quaternary locally repairable codes with code length n≥k+1 and small locality, and can verify quaternary locally repairable codes to attain SingletonLike bound or CadambeMazumdar bound and prove that except for 55 quaternary locally repairable codes, other quaternary locally repairable codes are all locality optimal.