To make something like this work it needs an angle. Greenscreen is the best approach as mentioned with a selection of backgrounds. You could have something like the eiffel tower, Pyramids, Standing on the moon or things that they very obviously can't do themselves and are fun and entertaining.
I'd put up signage indicating they choose their Background BEFORE they are photographed as it will allow you to best position them for some shots but more importantly, in case you do get busy you don't want people standing there procrastinating about which BG they want and wasting your time.
As for onsite printing being slow or difficult, Why the hell do people always think that?
I do onsite green screen setups and printing for corporate and Charity events and shoot 700 people in an evening . We usually do in excess of 400 6x8" prints with extras for groups and people who double up with friends etc.
Sometimes my wife will do crowd control/ direction ( which you wouldn't need) and then it's just my son and I. I shoot and he prints. Never had a problem keeping up yet,It's neither difficult nor time consuming
The GS Software composites the pics in seconds and you send the pics to the printers. You pre ajust the GS software so you rarely if ever have to ajust it thereafter and its a matter of import the pic, position the image of the people and print it.
If we want to do a pic package I have them setup in PS so we import the image, hit the preset action key for the layout desired and the thing spits out the printer.
In honesty, I think you are going to have to come up with something pretty out of the box to get run off your feet so I wouldn't be worrying about keeping up. It would take a line up of extremely unlikely proportions for this to happen -IF- you are set up right in the first place.
I usually take 3 printers to a job and only take 2 into the venue. Had things bugger up where one went down so we have done them on one printer only. I recommend taking 2 machines however and them printer pool them. This way if you get asked for 2 prints ( and I'd Include that as a value added option) then you can print both copies in seconds more than it takes to print one.
As for what you offer, I have a hatred for 6x4" prints and it defies my understanding why professionals even offer them. People here carry on about never letting their files out in case a client gets a bad print made yet they will happily sell a print size that is as amaterurish, least profitable and hides as much detail in the image as possible.
I have always done 5x7"'s as a minimum size and now I am doing 6x8's. Because of the amount of prints I crank out these days with my regular work I have got a fantastic deal on 8x12" paper which I buy at 3000 sheets at a time and then cut it in half for the event work. It's much cheaper than me buying pre-cut 5x7 and doesnt take long to cut up a couple of boxes to cover one of these events. Anything left over just goes back in the kits for next time.
Since doing the 6x8's we get so many more comments on the larger print size and although I put the price up $5, people are telling us what good value it is. I thought I was pushing the price but anyway... 
You have to realise this is a numbers game. I would NOT count on low prices bringing in hoards of people. In my lengthy experience this doesn't happen. You have to divide what you want to make by the price to see how many customers you are going to need. If you want $1000 ( which I would) and you are selling at say $10 per pic, then you are going to need 100 customers.
Divide that over the time the fair runs, ( 10 hours per day?) then you are going to need 10 customers per hour ( 1 every 6 min). Firstly thats easily doable production wise, 2ndly that sounds reasonable to pull the people in.
IF you go to say $5 per print, the dynamics change and suddenly you need 200 clients. That sounds like a BIG ask to me just to get them in. The low price is not going to double the amount of people that want what your doing that's for certain.
Plus you are going to be run off your feet and people will probably walk away.
IF you go to say $15 per print then the Dynamics change a whole load more favourably the way you want.
You will loose very few people with the extra cost particularly if your samples and presentation is right ( and there is no way you are going to loose 50% of those that would have bought at $10) and you have a lot better chance f pulling in the lowered number of clients that the higher.
And that's what I'd be worrying about, getting enough people not worrying about getting too many. If you are meeting your sales target and then can't keep up and people walk off but you still did double what your goal was, who cares? If you go in thinking you are going to get run off your feet and are not, then you'll be dissapointed.
Myself, I'd be going for a $15 base and then having options for upselling. My base package for my main work is $20. My average SALE value is $32.50. It's the $35 and $50 sales we regularly do with the add ons and options that really make me money.
I'd be doing a multiple print option ( 2 prints on an 8x12 sheet is real easy if your using 6x8s to start with) and I'd also do multiple backgrounds of one pose. In that case I would offer 4x 6x4's all on the one 8x12 like the famous Andy Warhol image.
You might also look at a picture package like 2 x5x7's and some wallets on an 8x12 sheet.
This is all very easy to set up with one printer loaded with 6x8 paper and another with 8x12 and preset the actions in PS or your greenscreen program.
Something you need to do with this and I would go so far as to say much of your success in this event will depend on it is SIGNAGE. You need to make it plenty clear what you are doing and attract attention with it. Make sure you do an overhead banner of some sort and have that up high where it's not going to be blocked by people standing in front of it etc.
If the fair goes into the night, make sure you have the banner lit up like Christmas. I'd also have some bassy music playing with upbeat tunes. You don't have it loud, but the bass will carry and pull people in even over the rest of the noise. Lighting and Music are my 2 secret weapons when I do trade shows and they allow me to buy the cheapest/ most undesirable stand at the event and pull more people than the guys right up the front that paid 3-4 times more.
One more idea that just came to mind is the possibility of something like a petting zoo you take pics of kids and families with. If you can get some lambs or baby Chickens, Piglets, ducklings etc and set up a scene with them and take pics, maybe that could work if there is no opportunity for the people to do it else where. I don't know, never been to one of these things, just a an idea I'm throwing out.
From RDKirk: First, let me check the forum heading...yes, it does say "Business of Photography" and not "Hobby of Photography." Okay. So we're talking about making money, not about hobbies. By "business" I am presuming activities that pay expenses and produce a profit over the long term.