In Greek mythology, Penelope was the wife of Odysseus. Odysseus was away from Penelope for twenty years. While he was away, Penelope was approached by many suitors, but she refused them all and remained faithful to her husband.
The speaker of the poem "An Ancient Gesture" is also a faithful and patient wife. In the first stanza, the speaker describes wiping her eyes "on the corner of [her] apron" and not knowing where her husband is. She described "burst[ing] into tears" and not knowing what to do. The image evoked here is of a faithful housewife whose husband is absent; the housewife is lonely and misses her husband. The image of the "apron" is significant because it places the speaker in the kitchen, which in turn perhaps makes us question the old sexist stereotype that said women "belonged" in the kitchen. This seems to be, perhaps, what the speaker's husband expects of her.
In the second stanza the speaker consoles herself by drawing upon the example of Odysseus's wife, Penelope. The speaker says that the kind of patience and faithfulness that she is showing is "an ancient gesture, authentic, antique." The logic here seems to be that the wife's faithfulness and patience are honorable because it has long been the expectation of women to demonstrate these virtues. This logic to a modern reader might seem rather flawed. A modern reader might question whether it is still right (assuming of course that it ever was right) to expect women to simply wait patiently for their husbands, even when they do not know where those husbands might be or what they might be doing. Indeed, making the readers question these expectations of women seems to be the main point of the poem.