WorrySighWorrySigh sorry to hear you are having such a bad start. I do think orgs that put time into proper inductions reap the rewards with happier employees. They sound very disorganised & unsure of what they want - it may be that you'll never be able to satisfy them if they don't know what they want.
I think the main question is if you quit what are the negatives? It is "just" a case of the money, or will you feel like you didn't stick at it? To me it sounds like you could probably 'fudge' the gap on your CV, so I wouldn't focus on that being an issue.
I had a difficult 6 months in a job, but I am glad I stuck it out as when I left I knew I'd tried my best and basically it was them not me! However, I too had periods of crying and telling DH through tears that I was concerned about getting a terrible illness (eg cancer) due to the stress. I left towards the end of my 6 month probation. I am still in touch with some from there and they are still bumbling along not knowing what they want 
.
If you have been a contractor before, could you adopt a bit more (am making lots of asumptions here) of a 'freelance' mindset. Throw it back to them; "Ok thanks for that feedback, can you tell me what I should priortise/do on x next time?" and then finish any type of performance meeting with a run through of the points agreed, maybe follow up on email.
When a new task is presented to you, appear at their desk with a notepad and go through the task? Or ask them discretely via email/in a room (maybe they aren't giving enough detail as they don't know the detail themselves?). Try not to be apologetic about this. They cannot expect you to magically know how to do x if they don't give clear instructions....
Try to think less like an employee but more like a service provider?
I started freelancing when I left the "horrible job" as it is known to my friends and family (!) and I sometimes do contracts and I find myself with a much more "You need to tell me what you want to focus on, I cannot do the 20 things you want done today" or "I am sorry, but I don't understand this email about x, what exactly should I do with it?" and not caring what they think of that!
If they aren't managing you well, you could try to manage them upwards? Also are there proactive things you can do for yourself (book induction meetings with others if it hasn't been done for you) to help? Worth trying to sure up what the job spec is meant to be and giving it to your line manager in draft "to help us both understand my role"?
However, I also think if it really isn't working and realistically you don't think it will in the long term and it is making you miserable then leaving can be an option.
Hope you manage to have a break from it all over Easter.
Good luck.