How were hamburgers served in the 1930s?

On a bun.

Although it is somewhat disputed as to who was the absolute first to serve a hamburger on the proverbial bun, Billy Ingram, the founder of White Castle, revolutionized the mass distribution of hamburgers by selling his with a bun starting in 1921 (cf. Oxford Companion to American Food & Drink, p. 65). Following the expansion of the White Castle franchise, other hamburger joints took off and, too, served burgers on the bun. 

The vision of hamburger patties served on a bun took wider circulation with the creation of the "Wimpy" character in the comic strip and cartoon Popeye the Sailor Man. The strip began running in the early twenties and became an animated cartoon in 1933. Wimpy's phrase "I'll pay you Tuesday for a hamburger today" appeared in the strip in 1932. 

In the 1936 film version of the play, hamburgers are served on buns, not kaiser rolls. (For whatever reason, the grab application won't let me take a photo from the DVD player).