Thank you, thank you, thank you for posting that solution! I don't recall cancelling a download so I'm not sure how my Downloads got disabled, but enabling them again did the trick! And I had already searched online a few different times and found no solution, so getting to this page was a godsend.
I'd like to just offer this as a point of clarification for some users, as the description of the process given in the solution didn't exactly match what I found in my phone's software, so there may be others out there who will be wondering: I'm on version 4.1.1 on my Samsung Galaxy S3...once in the Application Manager, I had to flick the menu bar at the top to the left in order to find the ALL tab. There was nothing called Download Manager...but I did as the solution suggested, and scrolled all the way to the bottom, even though everything in this list was alphabetical. And there, at the very bottom, was an entry that just read "Downloads". Next to it, it said "Disabled". I clicked that and clicked the Enable button. And that fixed my phone. I went back into this same list, and Downloads was now listed alphabetically.
Oh, and just to offer some of the symptoms I was getting because of this issue: I could not even open the Google Play Market...it would attempt to open, and then shut down immediately. I could view but not save any attachments received in email or with text. Plus, the error message popped up at random times as well, I assume because there were some automatic downloads that my phone was unable to receive.
Again, thanks for the solution!! You rock.