Beds were certainly used on stage in some scenes: the opening stage direction to the Folio edition of Othello, 5.2, reads "Enter Othello, and Desdemona in her bed."
Other large structures may have been used, as in the woodcut of The Spanish Tragedy, where Horatio is shown hanged in an "arbour." Realistic hanging scenes were arranged by the special effects department. Henslowe records a payment "for pulleys & workmanship for to hang Absolom."
A similar structure could have been used in Much Ado for the scenes of overhearing in a "pleached arbour" like the one illustrated here.