The phrase "education of the private house" refers to the daughters of educated men. Read more here The Rage of Virginia Wolf
How can we explain that amazing outburst in August 1914, when the daughters of educated men rushed into hospitals, drove lorries, worked in firelds and munitions factories, and used all their immense stores of charm to persudade young men that to fight was heroic? So profound was (woman's) unconscious loathing for the education of the private house that she would undertake any task, however menial, exercise any fascination, however fatal, that enabled her to escape. Thus consciously she desired "our splendid Empire", unconsciously she desired our splendid war.
I don't understand the underlined part. What does it mean?
What about fascination? What does it mean here?
It means anything that arouses her interest or curiosity.