"Socially constructed"

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I've been thinking and reading about the idea that sex and/or gender are "socially constructed."  This is often asserted by feminists who have a debunking and liberatory agenda.  The idea is that sex and gender "binaries" are not written into the nature of things, but results of choices, perceptions, customs, cultural assumptions, etc. You couldn't abandon the cat vs. dog distinction (it's real), but you could abandon the man vs. woman distinction (it's constructed). 

I don't quite get this, because "socially constructed" categories can earn their keep, even if they're not written into the natural world.  In fact they're quite diverse, coming to be in many different ways.  Some examples:

Doctors and lawyers.  Nobody's by nature a doctor or a lawyer.  As a society we created the institutions of medicine and law and established procedures that make one qualify, or not qualify, as a doctor (or lawyer). 

Teenagers, seniors. We have lots of age categories that are socially constructed.  To be a teenager, you have to have an age ending with "teen" but we draw a circle around people with those ages because of various facts about them, and also because of various decisions, perceptions, norms, etc.    Likewise, seniors have to be at the elderly end of the age spectrum, but there are further facts about them, and decisions and perceptions, involved in marking them out.  Seniors are not just old, but assumed to be leisured, retired, slowed down, etc. Other age categories are worth thinking about here too: baby, toddler, tween, middle aged, etc.

Blonds, brunettes, redheads.  If you listen to people talk about "blonds" you'll realize that a blond isn't just someone with blond hair.  Being "a blond" involves a certain amount of ditziness, extra attractiveness, and so on.  You can have blond hair but not be "a blond" and you can have brown hair and make yourself a "blond"--by dyeing your hair and taking on the necessary ditziness and sex appeal. Likewise for brunette and redhead--hair color is involved, but also character traits.  The whole thing's  bound up with culturally perpetuated perceptions.

These three examples reveal various things about "social construction":

1. The doctor/lawyer example makes it especially obvious that a category isn't disposable just because it's socially constructed.
2. A socially constructed category can have natural prerequisites. You cannot be a teenager without having an age in the teens.  So it's possible to agree that male/female is a socially constructed distinction but still think there are natural prerequisites for being one or the other.
3. The examples show that socially constructed categories vary in their superficiality and connection with mere perceptions. "Blond" is like that, but teenager much less so and doctor/lawyer not at all,

My sense (backed with no defense at the moment) is that biological sex really is natural, not socially constructed, but gender in a broader sense is socially constructed.  The closest analogy, of the three above, is the age categories.  Doctor/lawyer is much more bound up with institutions than gender, and blond/brunette/redhead is more superficial and bound up with (stupid!) perceptions.  There's a "constructed" aspect to being a teenager, as well as a basis in reality, and I'd say the same thing about being a man or being a woman. Which is not to say gender categories are just like age categories--just as useful, just as relevant in all the same contexts.  It's the type and degree of constructedness that seems roughly the same.

What I'm really interested in at the moment is the idea of social construction. It doesn't seem as if lawyers is a socially constructed category in anything like the way blonds is, and blonds seems very different from teenagers.  So assertions about the socially constructed nature of gender need to come with clarification:  in what sense?  The idea is more or less radical depending on the answer.