Drives in new external enclosure keep disappearing/stop responding

I purchased this product () in which to place a set of old HDDs/SSDs as external storage for a server machine I'm setting up.

The server is a Dell Optiplex 7050 (specs below) that I got refurbished. It's been working great so far, but I needed storage for it.

Intel Corei7-7700 (Quad Core/8MB/8T/3.6GHz/65W); supports Windows 10/Linux

8GB (2x4GB) 2400MHz DDR4 Memory

AMD RadeonR7 450, 4GB, (DP/DP/SL-DVI-I)

Windows 10 Pro 64bit

When I plug the hard drive enclosure into the Optiplex, it works just fine...at first. I can do whatever with them. My goal was to consolidate their contents and organize the files, and I can do that for about 5 to 10 minutes, tops. Then, they vanish from Windows, and any transfers I had going or whatever just...fail.

The device doesn't power off or anything. The drives seem to continue clicking away for a bit.

Turning it off and on again works sometimes. The longer I wait, it seems, the longer I might be able to use them - which makes it seem to me like a heat issue, maybe? I realize this thing doesn't have great ventilation, but...

What could be the problem, and how might I solve it?