Controllers on Steam randomly stopped working mid-way through games.

Are you connected trough a Steam Link? Wired or wireless?

Sometimes a Steam Beta update temporarily breaks the controller configs, are you using the Steam beta updates? If so I'd try stepping out of that and trying again. (Settings -> Account -> Beta Participation -> Change)

Else maybe go to Settings -> Controller - General Controller settings and make sure Playstation/Xbox configuration support is still checkmarked

Are you still able to bring up the overlay with Shift+Tab or not?
I've found that with an unstable GPU - whether factory overclocked or manually overclocked - the Big Picture Overlay is susceptible to crashing without taking down the entire game.
Reducing the GPU clocks has fixed this on 3/3 systems for me so far (all NVIDIA GPUs).

Do you use third party controller remapper apps? (e.g., DS4WINDOWS)

If yes, I'd try uninstalling any of those controller apps first and then let the Steam client auto-detect the controller.

If you disable all of Steam's controller stuff does your Xbox controller work fine?

Try reinstalling drivers for your controllers.

In the last few weeks my DS4 has been getting randomly disabled by Windows when booting up, so I have to go to Device Manager and enable it to get it working properly again.

Not sure if that will help you out but it's worth a try I guess.

It's been happening with my XB1 controller as well and I have also disabled the Nvidia streaming functions too as has been recommended on the Steam forums.

I'll try disabling the Steam beta settings next.

Are you using a good USB cable?

My DS4 often would disconnect with some of the ones I had I home

I wonder if this is the same thing that was happening to me last week. I was playing Mass Effect with the Wii U Pro Controller (via Mayflash adapter in XInput mode) and it just stopped responding a few times, if I was moving around at the time it got stuck moving into that direction etc. I've never seen that before with this controller. Happened maybe five times in 20 odd hours? Every time I was able to fix it by turning the controller off and back on though.

Turning it off and back on doesn't even disconnect the controller (as long as the adapter is connected Windows thinks there's a controller present) so I was thinking it's just an issue with the controller.

Does the controller still work in Windows' controller settings when this happens to you? There should be a place where you can check that each button works etc. (unless they removed that in Windows 10).

Do you have a console you could try your controllers on?
Or have you tried switching to wireless?