Disc playback: is it *all* the discs that exhibit this behavior? If so, then it is likely the player, but it's odd that both a dedicated DVD player AND two different PCs. What kind of discs are they? DVD-R, DVD+R, DVD-RW? I don't know that it matters, but there may be inherent problems with one of the formats.
To test for corruption, I would ask how are the files stored on the DVDs? MP4? AVI? If you can get one of the discs to read long enough, you could try copying a file to your PC and then playing the video from your computer. If the video plays fine, then a) the data isn't corrupted, and b) you're narrowing down the players as the problem. It might be a daunting task to play however many videos you have, but if you're able to copy them all to your PC (even a disk at a time) and test playing them from there, you can confirm. That leads to:
Storage/Cloud - The best option may depend on how much data you have to store. Industry best practice follows the 3-2-1 principle. 3 backups. 2 on site, 1 off site. That may mean storing all the files on your computer (first backup), backing them up to an external drive (second backup), and then syncing to a cloud service (3rd/off site backup). For home users, that's probably overkill, but having 2 copies of your data is great. One local and then either an external drive or a cloud service.
Providers like Google and Microsoft are obviously the heavy hitters, but they don't give the free accounts a lot of data. I don't know what Google gives, but my free Microsoft account gives me 15 GB. In my case, I have an O365 subscription with Microsoft, so I have 1 TB of available space in OneDrive and that works for me. Additional storage is pretty cheap though, with 200 GB at $2 a month, going up in $2/200GB increments. $4/400 GB, $6/600 GB, $8/800 GB, and $10/1 TB for additional storage.
PCloud, GoogleDrive, OneDrive, DropBox, and Box regularly make the top lists for Cloud-based Storage. It really comes down to a) how much data you have, b) the price you need to pay to store that much data, and c) what services come with it to provide the value for your dollar.