Straub was born in Wiesensteig, into a family of sculptors. His father Johann George Straub and his brothers Philipp Jakob, Joseph, and Johann Georg Straub were also sculptors, as was his nephew Franz Xaver Messerschmidt. J. B. Straub studied in Munich with the court sculptor Gabriel Luidl and then went to Vienna, where he worked from 1726 to 1734.
In 1734 Straub returned to Munich. In 1737 he was appointed by Elector Karl Albrecht from Bavaria as the court sculptor. In the same year Straub married a daughter of the court engraver, Franz Xaver Späth.
Straub worked primarily in Upper Bavarian churches and monasteries, frequently alongside some of the greatest Baroque artists of the day: the architect Johann Michael Fischer, the painter Johann Baptist Zimmermann, the Asam Brothers, the Tyrolian painter Johann Jacob Zeiller, and the stuccoists Franz Xaver and Johann Michael Feuchtmayer, among others. Usually Straub's figures are carved in simple white, with very little gold trim.
Important works by Straub are in the Residenz in Munich as well as in Schloss Nymphenburg. His best-known church works are the altars of the monastic churches of Andechs and Schäftlarn as well as St. Michael's Church in Berg am Laim, a borough of Munich.