Can I Keep the Box Turtle I Just Found in a Field?
Can you? Technically yes. But the better question is "Should you?". And the answer is most likely no.
A box turtle you find outside is most likely wild. And wildlife should be left free whenever possible. A lot of states also have laws about what you are allowed to do with local wildlife. Usually you are supposed to leave it alone.
Keeping That Turtle Could Be Illegal
In some states it is actually illegal to take a turtle from the wild. Yes, this includes turtles you find wandering in the road. (You might not consider the road to be "wild," but in this case it is).
Some states only have rules about certain species of box turtle. Especially Ornates. Many states do not let you keep Ornate box turtles for any reason. Others make you get a license to keep one legally.
Box Turtles Have Feelings, Too
How would you feel if someone plucked you off the street and stuck you in a cage? Even a nice cage full of your favorite furniture and food? Of course, that assumes that the creatures that captured you know your favorites. Which is unlikely because you can't tell them. They don't know if you're a beer and pizza or champagne and caviar kind of person.
No, box turtles don't have the same brain power as us, but they can still feel on some level. And different turtles do have favorite foods and habits. And in the wild they know where to find these things. So a wild turtle will generally feel happiest living free in the wild.
So the turtle in the field? You should probably let him go on his way.
Wild Box Turtles May Carry Disease
Actually so can box turtles from pet stores. In fact, reptiles in general can carry bacteria, especially salmonella, on their skin. This is the same bacteria that can cause food poisoning if you eat undercooked chicken or raw eggs.
So unless you are careful, that cute box turtle could make you miserably sick.
Yes, plenty of people keep these turtles as pets without getting sick. But getting one on the spur-of-the-moment because you "found" it is not the best way to stay safe. Especially if you have a young child or elderly person at home. Or if you're pregnant. Salmonella can be especially dangerous for these groups of people.
You Can Help a Turtle in Danger
That said, if you find a box turtle in a situation where he's likely to get hurt or killed—like the middle of a road—you can move him to safety. Just make sure you do it safely. In other words, don't risk your own life or other people's lives darting through busy traffic.
If you can tell what direction the turtle is moving, take him in that direction. If you bring him back to where he came from, he will most likely just get right back into the road or whatever dangerous situation you found him in. There's a reason he's going where he's going, even if you don't know what it is.
But don't take it too far when deciding on a "safe" spot. If he's crossing a road, just take him to the other side of that road. Don't take him to the park across town or the nature preserve one village over. He'll probably just try to get home and may get hurt (or killed) trying. Believe it or not, turtles can and do live quite happily in cities. They have their own "home range," and that's where they want to stay. Home might just be the abandoned lot right next to where you found him!