www.makemkv.com

So my kids love DVDs but since they are kids, they damage the DVDs quite fast, basically in a week or two after purchase the DVD is not playable anymore. Due to this I want to "losslessly" convert all the still playable DVDs for storing on external HDD but in a way that they are playable on Sony PS4 Pro.

I tried ripping it with Handbrake in high quality but the ripped versions look quite bad on UHD TV (though they are good enough for tablet).

I don't care much about the file size as long as it is around the DVD size (4-6 GB) as the storage these days are cheap and I don't have that much DVDs anyway (around 20-30).
But what I care for - is quality and ability to extract the videos faster than for hours like in Handbrake.

What I tried so far is - I took one DVD (Frozen movie) and converted it with MakeMKV to MKV file.
I got 4.8 GB file that did not play in Windows Media Player but could be played in VLC without problem.
I copied the same MKV to external HDD and tried playing in PS4 Pro but it said the file format is not supported.

Since I am quite new to video encoding, I don't know how to get around this issue. I searched and tried some programs but seems all have some cons - either they can't break DVD protection, or they can't extract the video (just rip) or the output is not playable on PS4 Pro.

As far as I understand the PS4 can play only MKVs that contain MPEG4 video but my version probably has MKV with MPEG2 inside.
I thought that if I could extract the MPEG2 video from MKV and just save it as MPG file, the PS4 might play it?

Any ideas, help is much appreciated.

The problem is, of course, limitations on the player software. Tools like handbrake (from handbrake.fr) can convert from MKV to device-tuned MP4 files, including ones that the PS4 will be able to use.

There are also tools like Plex that can act as media servers that will convert the files "on the fly" for supported devices.

I don't let my family have access to optical media - everything is immediately ripped, and anything that is going to be watched by "lesser devices" (i.e., can't handle MKV files) gets encoded as MP4.

Just to elaborate on what Woodstock said - in case you didn't already know:

The 'MKV' file format that comes out of MakeMKV works fine with VLC or Kodi media software and I guess Plex, though I've never used Plex. The 'MP4' file format is supported on more kinds of devices, Windows Media Player, and so forth. But the audio format and subtitle format from Blu Rays is not supported by MP4, so you need some other tool to convert a ripped Blu Ray from mkv to mp4 because it has to convert the audio and subtitles from the Blu Ray format that mkv can handle to one that mp4 can store. Handbrake and ffmpeg can convert mkv to mp4 and switch the Blu Ray audio, but I think you would need something else to convert the subtitles. Since you're talking about kids' films, I'm guessing subtitles aren't important anyway.

For me, I found that even if I stay with the MKV format it's helpful to convert re-encode the video and to convert the audio from the original Blu Ray format to the AAC format in DVDs. For watching movies on VLC on computers it doesn't matter. But I have the Kodi media software on a Raspberry Pi 2 by my television and it can't play plain Blu Ray rips from MakeMKV. But after a re-encode, it can handle them.