I'm also in Ohio and have kept quite a few turtles outdoors. In my experience heat is the biggest concern, and from the looks of your balcony there is plenty of shade. Assuming that turns out to be true, you should be able to build a decent enclosure and keep your pet outside. As others have mentioned, I would keep a close eye on the sun's position and the temperatures on your balcony. An unprotected balcony can easily climb over 120 degrees in many places, which will cook the majority of reptiles fairly quickly. Cold is generally less of an issue outside of winter- you can (and should) provide a sheltered area with a heat source. Keep this area out of the wind and pile it with substrate the animal can dig into. Turtles and tortises will generally figure out where the warm and cool areas of their habitats are and will move themsevles accordingly. You will have to winter the animal elsewhere of course, as attempting to hibernate it on the patio will probably kill it.
As for enclosure materials, you have a lot of options. Depending on how much space you are willing to sacrifice you will probably find regular plywood to be the most economical solution. You will want to make sure you coat it so that it doesn't deteroriate, but you also need a material that is safe for your pet. I recommend the two-part epoxy used by DIY aquarium builders, as it is both completly waterproof and non-toxic. Make sure whatever you build has sufficient drainage!
For substrate I would use cypress mulch- I've had good luck using it with all manner of reptiles, including box turtles. It's cheap, readily available, holds just the right amount of moisture, and is easy to work with.
If you plan ahead, you can easily turn the entire balcony area into a suitable habitat- adding some appropriate plants will not only make the area look better, it will make your turtle or tortise much happier. I suggest using the heavy-duty plastic pots for any plants, as you need something sturdy yet lightweight during routine cleaning.
Good luck