As whuber says, the short answer is that quota samples are the "poster child for outmoded, known-bad sampling methods" and "have long been discredited." The longer answer is that there may be conditions under which "quota-like" samples can work reasonably well.
Exhibit A here is recent work on reconstructing representative results from opt-in Internet panels. This paper gives the statistical grounding for this approach. To make a long story short, typical sampling schemes 1) draw a random sample, 2) attempt to recruit subjects, and then 3) add post-stratification weights to compensate for differences in who responds. In the opt-in approach, you 1) recruit subjects non-randomly, 2) compare responses to a representative baseline, and 3) add weights to compensate for the differences.
In terms of practice, opt-in sampling is similar to quota sampling, but the statistical foundation is more developed. The upside is that you can make claims about representative sampling, confidence intervals, etc. The downside is that your claims are based on difficult-to-verify assumptions about how people self-select into your sample.
A lot of people are skeptical about these methods -- they sound too much like quota sampling. But some evidence suggests that opt-in sampling can work well at least some of the time. So despite the controversy, Polimetrix/YouGov (an early adopter of the opt-in sampling model) seems to be doing reasonably well. Among other things, they've done all the data collection for the Cooperative Congressional Election Study, a series of recent academic U.S. national election studies.
(I'm pretty sure ICPSR carries this data. If not, Harvard's social science dataverse certainly does. Lots of academics are using data from these samples.)
Anyway, you asked about quota sampling. As you can see already in the comment thread here, any well-trained pollster will tell you that quota sampling is bunk. The jury is still out on opt-in sampling. For the time being, if you want to draw confidence intervals around quota samples, I'd say these methods are your best bet.