Dress
Oh yes women did! *LOL*
Nurses, maids, nannies, governesses, factory girls, the lot all wore skirts and corsets, long as it was in fashion. Heck some even wore huge hooped skirts or puffed up skirts when they were in fashion as well.
It was deemed "improper" for a woman to go about without her corsets, much like it would be today for a woman to go around bra less (or without a girdle in the 1950's though 1960's).
In the first place, dresses and bodices were fitted onto corseted bodies, therefore nothing would fit properly without those foundation garments.
Next, for saftey and hygine reasons, one would think shorter skirts, and or even better trousers were much better, but neither was really possible then. Decent women did not show their legs, much less ankles, regardles of their occupations outside the home. Trousers were the domain of men well until the 1930's or so. Even when riding habits changed over to breeches for women, some sort of skirt contraption was still required not only for modesty, but many hunting clubs would not allow a woman to enter wearing "trousers".
In many cases of nurses, maids and such, one wore what was provided, and those dresses were normally based upon the fashions of the day. Interestingly enough, women were often hired for some of those positions (usually maids), because they "fit the uniform".
As stated before, it must have been hell on earth wearing the several pounds of undergarments of that era, plus a corset in all that heat and heavy work, but then again many housewives wore the same things at home for doing everything from housework (including laundry), to farm work.
PBS had a great program several years ago called "1900 House", where a modern family "goes back in time" to live as one would have in Edwardian era. The mother and oldest daughter were put into corsets and promptly became short of breath. The mother in particular having to do all that hard housework wearing a corset. She tried to rebel once and not wear the thing, but then her clothing didn't fit, so it was back into stays she went for the duration.