Each DVD holds about 5Gb to 16Gb (depending on # of layers and # of sides). So, worst case, 60 dvds per Terabyte. Buy a HDD for under $100/4Tb and you can store 240 dvds (or more).
If you want to get fancy, a 4Tb SSD is about $450.
Then copy the DVDs to the useful, modern storage device. Then throw the drive and dvds in the trash.
I just installed a new hard disk on a computer connected to a Microsoft OneDrive. When I reconnected to my OneDrive, everything was synchronized automatically - I was almost disappointed I didn't get a chance to clean things up and arrange them differently this time. In the end, I was super happy I didn't wast a half a day trying to copy everything back onto the New PC (except for a few applications).