It's not clear what year Baber was born in. His World War I Draft Card shows 1888, though his World War II Draft card shows 1890. Further, Baber's tombstone gives 1892.
Baber attended and played baseball at the University of Arkansas.
Baber played with the 1913 Texarkana Tigers.
Baber died in 1960 and was buried Duncan Cemetery in Waldron, AR.
"Sherman, Tx., June 10th. --Pitcher Baber, who has been with the Sherman Texas-Oklahoma League team, left for Muskogee, Ok., having been sold to that team in the Western Association."1
"Two former pitchers of the University of Arkansas are on T.O. reserve lists this season. They are "Hatch" Black of Little Rock, with Sherman, and Oscar Baber of Texarkana."2
Statistics at Baseball-Reference.com.