But legally it is difficult I think because they are private clubs.
It's not difficult. They are still institutions given status under the law and enjoying the protection of the law. We could as easily apply equality law to them (as, I believe, we do in regards to race) as we do to privately owned companies.
The barber's thing wasn't a matter of skill, she was asking for what would normally be considered a man's haircut. But as I say, I'm not sure if that refusal was supported in law. It may be that it wasn't.
School - yes complicated. But the reality is you don't have to justify a single sex school by showing how it will make things more equal - you can just do it. And that's problematic, I think.
And that's more my point - I don't support single sex spaces just "because". I think they ought to have a justification (and organizing against the patriarchy is very much a justification, but "I like to play with kittens and I don't want women/men there", probably isn't!).
What I fail to understand at all is how single gender spaces in places that do have to uphold equality law can be supported under the current law. They aren't sex segregated, which can be allowed when justified, but they impact against sexes in an uneven way (an all male gender space will exclude a few men but almost all women). So they are discriminatory. Any lawyers explain how that fits?