I don't think this is such a terrible thing, although clearly this adversarial approach to testing ideas isn't for you, or everyone. Enrico Fermi used to challenge grad students with similar questions, the famous example being 'how many piano tuners in Chicago?'.

I don't think it's a stupid or impossible question, but one that seeks to establish how you go about estimating tricky things and thus identifying fruitful territory for more rigorous research. Let's walk through it, with the awareness that we're trying to refine our techniques of estimation rather than caring about the actual number of barbers.

'Barber' implies haircuts for men. This is good, because the number of hairstylists that cater to women depends on fashion, which is hard to quantify. On the other hand, hair cutting is more related to hair growth. There's about 3/4 of a million people in San Francisco. Roughly half of them are men, so we have 375,000 potential customers. Most men wear their hair short, off the ears. Of course some wear their hair long and have it cut very infrequently, while others like it much shorter, eg with a buzz cut and have it cut more often...but some of them may cut their own hair, since a buzz-cut isn't too challenging. I'm going to assume the number of hippies cancels out the number of close-cutters because San Francisco has a bit of a hippy reputation. So much for hairstyling.

OK, so I've postulated 375,000 men who average out to a 'regular hairstyle'. And I've postulated that a regular hairstyle means not having your ears covered, ie the old 'short back and sides'. While it's growing, you could comb it out of the way or let it cover the tops of your ears, but once it gets half-way down then you're either growing it out or it's time to visit the barber. How much hair is that? My ears are about 2 inches high, but I have small ears. 3 inches sounds pretty big though...I'll guess the average is 2.5 inches, so half of that is 1.25 inches. That's already pretty fluffy though...let's round it down to 1 inch. OK, so 375,000 men get a haircut whenever their hair has grown by more than an inch.

How long does that take? Hair grows at about 4 inches a year, so let's say 3 months. Which gives us 1.5 million male haircuts a year for the whole city. (we could have started calling the population p and the rate of hair growth g[h] and so on, but why bother, it's a chain of reasoning we're concerned with here). A haircut takes, what, 20 minutes? it could be faster, then again some want a more difficult style. So let's guess that there's half a million hours of actual hair cutting (including the bit with the sheet and the sweeping up etc) going on in a year in SF.

Now we're getting somewhere. How much does a barber work? I'd say 45 hours a week. Some barbershops open early for the business crowd, but then they're closed on Saturdays. Monday-Saturday seems reasonable, as does 2 weeks' holiday. So one barber could spend a maximum of 2500 hours per year cutting hair. Only, that seems a bit high. Sometimes you wait in line, but I don't usually think of barbers as busy flat-out most of the time. 30 hours of cutting and 15 hours a week of of waiting for a customer seems more realistic. So 1500 hours of haircutting per barber (edit: 1650, but who cares). And dividing that into half a million hours of cuttable male hair per year gives me 333, the number of people needed to cut all that hair.

At this point you can start arguing about whether someone that caters to both men and women is truly a barber, or whether women can be said to be barbers even if they only cut men's hair. If your answers to those questions are no and no, and we go with the stereotype of a barber as a man who only cuts other men's hair, then I'd halve these numbers and guess that the number of 'pure' barbers in SF is 75-100.

OK, so this is 10 minutes of my life I won't get back, but if I was going to market something specific to barbers then I now have a decent guess at the number of potential customers in my local metro area.

Of course, you could have just said 'pass me a copy of the Yellow Pages and I'll tell you'. So having done the above, I went and looked that up and got 131 results at http://www.superpages.com/yellowpages/C-Barbers/S-CA/T-San+F... ...with 2 or 3 cutters at each shop, that would suggest somewhere in the 3-400 range, but some of these entries are for Salons and some are in Daly city or Sausalito, not quite in San Francisco. Still not definitive, but not too far off either. Sure, there's a lot of guessing and hand-waving here...but the important thing is that I arrived at an estimate that roughly correlates with reality. In this case I was able to check that reality (by looking up the number of self-identified barbering businesses) but once you become adept at this technique you can use it for anything, and then get on with solving the problem instead of feeling stuck by a lack of information.