Playstation VR ($399; reviewed here) but only if you have a PS4 Pro ($399), the newer, more powerful version of the PS4.
You can now watch 360 videos stored on a USB or other drive using a Playstation VR. However, the resolution of the 360 video was limited to 1920 x 960. The PSVR's screen has a resolution of "only" 1920 x 1080, but because you're only viewing a crop of a 360 video at any given time, the PSVR's screen resolution is theoretically capable of displaying as much as 8k resolution (see the equivalence chart here). A full HD 360 video would only be similar to a 480p video, far below the PSVR's screen resolution limit.
With the update to the PS4 Pro, you'll now be able to watch 4k 360 videos, which will have detail that can be similar a 720p video. Of course the actual level of detail will depend on the camera(s) used (not all 360 cameras are the same image quality even if they have the same nominal resolution) and variables such as the lighting conditions.
But before you decide to get a PS4 Pro and Playstation VR, you ought to consider getting a VR-capable PC instead, which can now be had for about the same price as a PS4 Pro.