Cube projected image texture as HDRI?

I don't think you can use an image like that as an HDRI, however, you can convert it into an equi-rectangular map that can be used.

Give a basic cube your Cube-map texture (it's default UV is already in the right shape), and align it properly in the UV editing tab. Make sure to connect the image through an emission shader (so it becomes it's own light).

Then, select your camera and place it in the center of the cube (since I didn't move the cube at all, it is still sitting at the origin, so Shift + S > Cursor to world origin, and then Shift + S > Selection to cursor (on camera)). I scaled the cube slightly (in edit mode) to make sure it is bigger than the camera. Press N to bring up the transform menu for the camera, and set it's rotation to 90 on the X axis, and 0 on the others. Make sure any lights are removed from the scene, or moved out of view.

Then, making sure you're in cycles, select the camera, and go to the camera properties panel. Change the lens type to Panorama, and the Panorama type to Equirectangular.

Under your render output settings, make sure that the width is double the height, whatever resolution you choose. I put 2048 X 1024, which I think is the very minimum you should use to avoid blurriness. As per convention, powers of 2 are always preferred.

Then, you can render the image - you can even turn the samples down because the camera is very close and is receiving light directly from the emission. After the render, save the image, and there you have it - an equi-rectangular version of your image. You can then load it as an environment texture and connect it as an HDRI as per usual.

Here it is connected as an HDRI. The image is blurry because for my base image I just used a cheap low-res image I took from a google search. This demonstrates, however, that the resolution of the base image matters to an extent.

In addition to the emission cube I started with, I added 2 additional objects to the scene - a rough cube and a shiny ball, just to demonstrate that it gives proper light and reflective contributions to the scene.