I get an occasional problem where I can't save at any point between missions. The Save option is blacked out .

It also doesn't autosave when this happens. Continueing to play doesn't help because the Save option simply doesn't open back up.

Annnd, I can't quit. It pops up Quit or Cancel , I hit Quit and it just sits there until I reboot.

Tonight, I beat a mission I've been trying for days, only to find i couldn't save. Very frustrating.

Before and after missions, at payday and critical events the game make a autosave.
You can make manual save at any time, expect during unsolved events and fights.

If this isn't the case for you...
I. Your installation might be corrupted
II.You have to many savegame, many players report that saving/reloading works worse the more savegames you have
III.There's a rare bug that set the savefolder for read only or blocked, check the folder settings
IV. If you use galaxy that might cause your problems esp. if you activated cloudsave.
V.Savegame location&rights managments, several player reported problems with access rights and the usage of different storage mediums for game and savegames.

So..no real solution but some reasons that are known to cause similar problems, good luck.