Women in the 20th century in the Western world were not "barred from the public sphere in general" LOL. At all. Women were allowed in a vast number of spheres and spaces. Women could even be US Congress members and US Senators, state legislators, governors, mayors, civic leaders. In the 20th century, India, Israel, the UK and Pakistan all had female prime ministers.

But for the first three-quarters of the 20th century, the vast majority of women in the Western world were barred from many specific jobs and professions because of our sex. There were plenty of women working in all sorts of factories, offices and business enterprises throughout the 20th century - but only in certain jobs (except for during WW2). Women weren't kept at home chained to the stove, barefoot and pregnant throughout - women were allowed to participate in the larger world outside the home, but with a few exceptions like Indira Gandhi and Golda Meir, only if we kept to our place. Women could work in business, but mainly as secretaries and file clerks. Women could work in restaurants, but almost always only as waitresses, not as cooks or chefs. Women could work in banks, but as tellers, not as loan officers and executives...

In the 19th century, women responded to being barred from male institutions by setting up some female-only analogous ones such as women's colleges and universities, some as sister schools to the male ones (Radcliffe, Pembroke, Barnard), as well as medical schools, and of course women's hospitals and health clinics. But for the most part, this was not the case in the 20th century. When women were allowed into areas that previously we'd been barred from in the 20th century, particularly in the 1960s and onwards, we did not seek or get our own sex-specific institutions in general - except that is, for facilities like rape refuges, DV shelters, consciousness raising groups, support groups, rehab programs. Women who started attending Yale in the late 60s didn't seek or set up a Yale 4 Women. Women who entered various professions like law, medicine & business in unprecedented numbers in the 1970s did not set up women's law, medical and business schools - or aim for separatism in the world of work after graduation. Women who fought for the right to run in marathons did not seek to have a separate NY Marathon and Boston Marathon for women - in fact, when the NY marathon tried to segregate women, all the female runners protested and refused.

Where existing institutions already existed and sex mattered, such as in scholastic sports & dormitory provisions, women in the 20th century set up female-specific sports programs, places (locker rooms, loos, dorm rooms or sections of dorms) and social spaces (sororities, women's centers) within the extant institutions - but by and large, we did not advocate for creating a whole bunch of entirely separate institutions. We wanted our own locker rooms and sports competitions, but no girls & women in JHS, HSs, colleges & universities were advocating for our own female-only gymnasiums, field houses, ball fields, track & field facilities, courts, rinks, pools, boathouses and such. And where sex did not matter - such as in classrooms, libraries and labs, we wanted to learn and live in a mixed sex environment. In many previously all-male universities, women didn't even want separate housing - we preferred to to live in mixed-sex dorms where sex segregation only occurred by floor, section, pods or room, and where both sexes could come & go freely at all hours.