The Tango API is only available on Tango-enabled devices, which there aren't that many of. That being said, it is possible to create your own motion-tracking and depth-sensitive app with standard Java.
For motion-tracking all you need is a accelerometer and gyroscope, which most phones come equipped with nowadays as standard. All you basically then do is integrate those readings over time and you should have an idea of the device's position and orientation. Note that the accuracy will depend on your hardware and implementation, but be ready for it to be fairly inaccurate thanks to sensor drift and integration errors (see the answer here).
Depth-perception is more complex and would depend on your hardware setup. I'd recommend you look into the excellent OpenCV library which has Java bindings for you already and make sure you have a good grasp on the basics of computer vision (calibration, camera matrix, pinhole model, etc.). The first two answers in this SO question should get you started on how to go about determining depth using a single camera.