30x30 outdoor run with an 8x6 coop is plenty big enough for 50 birds. Personally, I would probably raise about 250 at once in that size area, but your back issue is definitely something to consider. Cleaning an area that is that dense with birds would be demanding to keep up with it in the last few weeks before butcher though. Which brings up another point. You want to size everything for your last three weeks before butcher. That is when the work will be the hardest. If they are in that big of an area starting from day one (or whenever you receive them) it might take a few weeks - maybe even a month for them to dirty that space, but as they grow bigger, eat more daily, and poop more daily, their pension for getting an area really dirty quickly will be obvious. I alwasy try to think of the worst bottle neck in the process and limit my flock size based on that factor - for me it's processing. I can only get about 60 or 70 birds loaded out of the pens, transported to my butchering space, and butchered in one day with help from one or two friends. So that number is where I have to draw the line (since I don't like to butcher multiple days in a row)

Stuff to think about
Cheers