I’m looking to get snow tires for my '78 Toyota pickup. I have 205/70/15 on there now. How big or small can I go with my snow tire purchase?
First, on every vehicle sold in the US – and I am told this is worldwide, but I have not been able to verify this - there is a sticker – commonly called the tire placard - that lists the original tire size and the proper pressure for that size. The placard is usually located on a doorpost or in the glove box – but sometimes it is located in the trunk or on the fuel filler door.
Tire Guides - a book that summarizes vehicle tire placards says there are 3 Toyota pickups produced in 1978: HiLux (6.00-14), SR5/SR4 (185R14), and what is identified as LongBed/Short Bed ( 7.00-14). So we are going to need to know what the placard says - and that will determine the smallest tire you can put on the vehicle.
But to determine the largest tire requires someone to have measured the clearances and - to my knowledge - no one has done so. You might do better to ask at web sites devoted to your vehicle. They are much more likely to have already done what you are considering.
Be advised also that going as large as possible with your winter tires does not necessarily make it better on bad roads. I had a '79 (longbed) and the rear ends are very light. Too much tire can ride on icy snow instead of biting into it and the traction be horrible.
I’d recommend sticking with the OEM size.
CapriRacer has listed the stock sizes, and they are all R14! So I think sticking to the OEM size is a boat that has long-since sailed.
I had a version of this truck, and not even snow tires will help much with the rear-end traction. You need to add weight over the rear axle to help with traction, like cinder blocks between the wheel wells. I put two 50 lbs bags of road salt between the wheel wells when the weather called for snow, and it helped A LOT! Bonus, if it got stuck, I could use some of the salt to help get out.
If you want snow tires, go with a 195-75R15, which was the stock size for my 1990 'Yoda. The tire diameter is very close to the same, it will fit the current rims, and a little more narrow will help with snow traction.
Agreed. I live in Seattle now, not exactly snow country, but have dealt with driving in parts of the country where there’s a LOT of snow regularly. My experience has been that tire size matters much less than tread pattern. Whether you use studs or not, a winter tire will have a different tread pattern than a summer tire or all-season. I don’t miss having to store extra tires for the other half of the year. 
Cinder blocks? I’d rather not put something that can slide around in the bed. Sand bags are a better bet. If you get stuck, you can even use the sand to gain traction. You can buy 50# bags at your hardware store. I’d get 3 or 4.