It's neither old fashioned or sexist by default, but it could be both or either. It's a personal comfort thing. Think of it this way, there are women only clubs, men only clubs, gay only clubs, lesbian only clubs, kids only clubs, elderly only organizations, etc... There are places you can only go if you are a Christian, Jew, Muslim, Buddhist, Hindu, sometimes on proper rank within any of these. Some places you can only go in certain positions or with X amount of income either high or low. All sorts of stuff out there hinged on isolation from other people.
The only way this would be sexist is if he were to not want to be alone with a woman because he distrusts women, or thinks they are lesser than men, or something along those lines.
Personal space is personal space, people have safe spaces in colleges.
While I don't understand the mentality, doesn't necessarily make it sexist or old fashioned, it's just how they roll.