I'm writing this from the Middle East where I moved (temporarily?) just over 3 years ago. Sadly, I'm in Saudi Arabia which isn't nearly as nice as Qatar as its far more extreme but a friend of ours who was here in Saudi for years moved to Qatar last year and said it is much better than Saudi, eg it is much more relaxed there and he can get an alcohol licence whereas Saudi is totally dry (although my son's chemistry teacher makes a quite decent red wine _
My DH was out here for 7 years before we met and when we got married it was a big decision as to whether to move me (and my two children then aged 12 and 9 out here) I also had a really good job in the UK which I we knew I couldn't continue in Saudi (not much call for female divorce lawyers in Saudi, they either say 'I divorce thee' three times or else get another wife. They can have up to 7 although my DH questions why any man would want more than 1 wife!)
anyway, enough rambling, it was the right choice for us. The pay is excellent (although not quite as good at the moment as the exchange rate is crap) DH does not pay tax, we have a free villa on a large UK compound with various facilities, ie indoor and outdoor pools, a sports centre, hairdressers, beautician, restaurants etc. We are quite a drive from the nearest mall (around 30 mins) and our lives are restricted somewhat, women can't drive and are third class citizens, I have to wear an abaya (big black number) and headscarf when out in public. Salah (prayer) is a pain in the arse as the shops/restaurants shut for prayer and its not uncommon to be in a restaurant and it will suddenly close and you are left sitting in the dark for 20 to 30 mins but you live with it. As with everything, there is the rough and the smooth.
At times we get very pissed off about the restrictions, the lack of bacon a glass of chablis, however, being here has enabled to pay huge chunks off our mortgage and provide a lifestyle for our children that we wouldn't have had otherwise.
We had decided to return to the UK this summer but given the current financial situation in the UK (and that my DH works in finance) we have decided to stay an extra 2 years to see my eldest DC through his A levels in a fantastic UK school. My DH's company pays us a school fees allowance which although it doesn't cover it, is the difference between DS going or not.
The schools out here are not bad. I'd say not as good as a private school in the UK but certainly as good if not better than some state schools. There are several British Schools in Saudi and I know it is the same in Dubai and Bahrain so I assume Qatar would be the same.
The important thing to consider is the ages of your children, and how long you think you would be out there. When my DS came out here he was 13 and we were therefore committed to him staying until he had finished his GCSE's which he will complete this summer before returning to the UK for his A Levels. We took the decision to send DD (aged 12) back to a UK boarding school last September as we didn't want to run the risk of her leaving here part way through her GCSE course.
What you need to research as well is where you would be living. If your DH would be working for a UK company then the chances are you would be living on a Western compound with lots of other expats. If that is the case, then you and the children would have a very good social life. If living in a villa not on a compound then that would be a little different and would require a little more consideration.
I would not write it off straight away. If your financial situation is stretched in the UK, then a few years here could set you up for the future.
We are leaving here in 2 years and will return to the UK owning two mortgage free properties, with a tidy sum in the bank and my DH who will be aged 45 then may not have to work. Its been a sacrifice at times, but well worth it I think.
Lots to consider. Why not have a look at the living overseas section. I know there are quite a few mums in Dubain and Bahrain but I think there is someone there in Qatar who may be able to advise you. Good luck x