Eliezer’s being “ruler over all that he had” means that he mastered the Torah of his teacher.
On the daf / in context: Yoma 28b presents Eliezer as a student who mastered Avraham’s Torah, continuing the motif of continuous yeshiva study.
Type of Learning
Custom (Bavli — Eliezer as yeshiva elder)
What Is Learned
“Ruling over all that he had” is read as mastery of the master’s Torah; Eliezer is therefore depicted as an elder who sat and taught in the yeshiva of Avraham.