How can I shoot a smooth 360° orbit around a stationary subject without a circular track?

A simple, practical method is to use a string as a radius guide.

Anchor one end at the center of the subject area (or peg it into the ground) and attach or hold the other end at your camera or tripod position. As you move, keep the string taut; that maintains a consistent distance and gives you a reliable circular path. If you don’t want the string physically attached to the camera, you can still use it to mark the track on the ground and walk that line.

For shooting consistency, use manual focus after confirming focus at the chosen distance, so the camera doesn’t refocus from shot to shot.

If the subject is small enough to rotate, an alternative is to place it on a turntable and rotate the subject instead of moving the camera. That can make distance, lighting, and background more consistent. But for a large, heavy subject like a motorcycle, the string-guided circle is the most practical approach.