The Wikipedia article on pearl hunting includes a photo of a diver with no mask in the water. So certainly some diving was done with bare eyes into the 20th century. However this article on Japanese and Korean pearl divers (Scientific American 1967) mentions a century of goggle/mask use. (Note, the article contains uncritical discussion of hyperventilation techniques - don't try this at home).
A more recent article (Gislén et al., Current Biology 2003) discusses vision in the Moken Sea Gypsy people of Southeast Asia, who live by foraging from the sea. Their children have far better vision underwater than European children. According to the article, they constrict their pupils more than Europeans underwater (ours dilate in response to the dim light). Like a pinhole camera this sharpens the image on the retina. Their eyes can also accommodate better underwater - i.e. they actually can focus better. While training from a very young age is clearly part of this (and would apply to the likes of pearl divers as well), the possibility of an evolutionary adaptation is also mentioned.
Goggles are older than we think though. Wikipedia cites The history of underwater exploration, Robert F. Marx 1990 for a claim of goggles made from polished tortoiseshell it 14th century Persia, and this unsourced article mentions a few other early goggles (such as wooden ones trapping air like a diving bell while the wearer faced downwards - you'd still get some lensing from the meniscus but not much. Once glass became available it was soon adopted.
Of course diving bells could also be used - to provide air both for breathing and as an interface to the eyes. These were mentioned by Aristotle in the 4th century BCE.
As for the irritation if goggles weren't worn, people can easily become accustomed to common forms of irritation, especially when necessary for survival. We sometimes forget this is our comfortable modern world. Indeed some people apparently (article discusses a more comfortable self-equalising system) use water-filled goggle with corrective lenses, to avoid needing to equalise the pressure.