Well, if good old e-mail does the job, why do you run a LMS? I don't want to manage assignment uploads of a class of 1000 manually, whether as e-mail attachments or in my local drive or in the "cloud". You should know, I personally never taught a class of 1000!
Talking of "cloud" the common opinion here is not that cloud will solve all your problems, rather https://docs.moodle.org/en/Performance_recommendations
. Also see the other information linked to the header of this forum.
Now to your specific question: Yes, a "crash" a possible. A decent server shouldn't crash, it will simply take very, very long so the users think it has crashed. Let's look at the possible weak points.
Let's assume that your server has a 1 Gbit/s (nominal) bandwidth, a practical limit in the range of 900 Mbit/s, which is decent. That gives 1 Mbit/s upload on average for your 800-1000 users, which is slow for the developed world. The slowest ADSL link to any POTS line is in the range 20 Mbit/s down, 4 Mbit/s up. At 1 Mbit/s 20-25 MB needs 160-200 seconds, roughly 3 min. So, if all of them start uploading 2 min before the dead-line, all of them will be cut-off by Moodle, even if the server can process that data stream.
The next question is, can the process the data stream, 900 Mbit/s, say, if the users start 3+ min before the dead line would the server be finished at cut-off? That you need to investigate. Generally, the data will be sent through the web server, which comes with a heavy overhead.
And finally, your storage, the "moodledata" must be capable of writing 1000 files arriving at 900 Mbit/s total. Again, mechanical drives will be different from software drives, virtualized drives will be different from physical drives.