Is there a general number of fields where people will likely be turned off and not even fill things out? When does frustration begin to set in in a long form with a lot of required fields? I understand if this question is too general.
It is a general question that can be answered with a general answer:
One more than is actually required by the business is too many.
In other words, make sure all the required fields are essential to allow the user to progress. All too often, the required fields are only required in the sense that someone on the business side wanted the field, rather than it being needed.
This post from Hubspot shows some interesting results for the number of form fields vs converstion: http://blog.hubspot.com/blog/tabid/6307/bid/6746/Which-Types-of-Form-Fields-Lower-Landing-Page-Conversions.aspx
They then (partially) break this down by input type.
It's interesting to note that conversion appears to go up with from 1 to 3 fields and doesn't go back below the conversion rate of a single field until 6 or greater. If I was guessing at the reason it might be either that customers perceive utra-simple forms to be trivial and so not worthy of filling in or it might be that the products that are asking for very few form fields to register are not attractive (trivial or beta splash pages?).
Another interesting find in that post is how badly select boxes affect conversion. My theory about this is that a customer's perception of the complexity of a form (rather than just length) is crucial in their decision to fill out the form or not and select boxes appear as a complex input type to users. AutomatedChaos pointed out I'd read the scale of that graph wrongly.
Anyway while the general principle of less-is-more almost certainly holds true for forms it would seem to be a takeaway from this that I wouldn't start getting overly worried until you have more than 6 fields.
"When does frustration set in?" This depends a lot on how the customer feels towards your brand and what their motivation is. If the goal of the form is to help the UN save cute kittens then they'll tolerate more work than if it's a customer feedback survey from Walmart. Equally high motivation will carry them through a long form - if they're trying to claim a winning lottery ticket for millions of £££ then you can make your form as long and painful as you like and whoever has the winning ticket will happily fill it in then thank you profusely.
This depends on what kind of information you're trying to gather.
In general, the correct number of required fields is the absolute minimum number of fields required to make a record usable.
If you are collecting information about a new user in your system, you may need only an email/username and a password. Without a username or password, a user record can't be used to identify a user and is useless.
If you are collecting information about a photo upload, you may only need the photo itself (no caption, tags, etc). Without a photo, a photo upload record has no meaning.
Also keep in mind that just because a field is "required" doesn't mean users will put meaningful data there.
Here is a good piece on the required marked vs. optional marked fields:
Marking the few optional field takes the onus of the user so much, and aesthetically having numerous red stars next to your fields does not make for an easy to absorb form.