Did DVDs exist back in 1987?

No. It was a CD Rom

ED-209s or RoboCops didn't exist either. So its a safe bet the story is set in the future.

CD. Duh.

A lot of people here seem to be missing the main point that the film wasn't supposed to take place in 1987 but a semi dystopian future around the year 2040, featuring lots of technology that didn't exist in real life. So with that in mind, a video disk certainly wasn't any sort of stretch, especially considering CD-Roms were already on the market by the 1980s.

It was a prediction made by the movie. At that time everything was on tapes.

As a former DVD lover, I got the reference, but it was quite cheesy how quick the disc loaded in the player and started.

Laserdiscs existed in 87. These are just futuristic, miniaturised versions. Video CD ROM was also around at the time.

From a technological perspective video playback from compact disc like media was possible at that time there were many solutions that date back to the creation of Laserdiscs ( launched in 1978 before the Audio Compact-Disc ). It just never hit the consumer market until CD-ROM (1988), CD-i (1990) and VideoCD (1993) utilized the smaller Compact-Disc.

If you really want to be surprised at some of the stuff that got predicted in films and TV lookup some of the older future house and future technology produced educational films from the 40s/50s/60s.