Just some things I recall from really bad games in the 90s and early 00s:
-Only having one or a few lives for the entire game
-No saves/continues (usually combined with the above)
-Ridiculously short timers for getting through an area
-No health meter (one hit and you're dead)
-Having to go back through areas of the game you already beat to do some inane mission that's slightly different and feels like it was added to pad the difficulty/play time
-Spawning for the sake of having more fodder thrown at you (or spawning in ridiculous areas like the ledge you're about to jump to and the enemy was not there before jumping). I still hate spawning to this day in most cases. If you just left an area and destroyed everything in sight, it should not be totally rebuilt as soon as you turn around.
-Unfairly slanting difficulty towards enemies (like bullets that travel through objects, while yours do not)
-Dying and having to restart a huge level all the way from the beginning
-Poor clipping so your weapon passes through enemies unscathed or you get hit by an enemy when you shouldn't have. Also clipping that results in you missing a platform when you clearly didn't.
-Poor controls (still a problem now, but nowhere as bad as it was)
My list is nowhere complete though and I'm sure others have plenty of annoyances to amend it with.