Frame Challenge: Modern Redundancy is WAY Better than you think
So let's say you manage to get a virus into Gmail or Outlook 365 ... did it wipe out all the email accounts?
No, you did not. Major email providers use multi-regional replication and distribution meaning that your email account exists on more than one server farm protected by more than one network. So, if a hacker, hurricane, or act of war wipes out a server farm in Texas, then all of your information still exists somewhere in at least 1-2 other locations in the world. So, let's say you live in Texas and go to log in and the system can't find your account there anymore, it will try to fail over to the next closest one. So, your download speeds might be marginally worse going through Washington or New York, but you probably won't even notice a problem.
Okay, so a normal virus is not good enough... what if you managed to upload viruses to ALL of the server farms at the same time. This is way easier said than done. Each server farm is a unique network with its own security measures and IT infrastructure in place. So, just because you figure out how to pernitrate one, does not mean you can get all of them... that is, unless you can find a vulnerability in their data replication software itself that ties them all together. So great! Now you can put a virus on ALL of their servers. You got them, right? ... right? ...
Not exactly, each one of those server farms is using offsite backups to various third party vendors. So even if you take down the whole distributed network, it is far from irretrievable. The email provider will need to spend a day or two identifying the vulnerability, patching it, wiping all their servers, and then reinstalling all their backup files. You will have an email outage for a few days, and maybe loose a couple of recent emails... but by in large, 99% of what you need will be recoverable.
Okay, but what if the virus includes a worm that can make its way into the backup servers? These do exist, but as I mentioned before, each server farm is a unique IT entity; so, if one server farm uses AWS backups and another farm uses Azure backups, then a worm designed to pernitrate one will not get the other; so, they can rebuild most people's accounts even if a few server farms also loose their backups.
Alright, gloves off! Let's pretend someone comes up with one of those TV trope super smart viruses that magically works on everything allowing it to hit all server farms and all backup farms. What most people don't understand about this trope is that the more malicious things you design a virus to do, the easier it is for antivirus software to detect and stop it. So, a virus that can exploit 10 vulnerably has 10 chances of being detected vs a simpler single function virus. This makes this trope annoyingly silly to anyone with a background in cyber security. But let's handwave away this issue and say you have one anyway.
Then... you've still only done minimal permeant damage. In addition to offsite automated backups, many companies also use local air-gap manual backups. These backups are not run as frequently as the automated off-sites, so you may lose a few days or even weeks of data, but most actual accounts (accept for those that were just created) could still be recovered because there would be no way for a virus to get onto them. Even if you get a dormant virus onto them, things like backup tapes have no way of rewriting themselves; so, you study the virus until you figure out how to safely remove it, then just copy the backup tapes running the data stream through an antivirus algorithm designed to filter out the virus as it comes out.
The only reason an entire massive sector of the email market would disappear is if a major corporation decided to discontinue email services. In this case, you would not suddenly loose your email, you would have months if not years of warnings letting you know to change providers, and chances are, some other email company would be more than happy to come along and buy those accounts and continue running them as their own moving forward in which case, nothing would really happen at all except for the usual bad customer service that happens during corporate mergers.
What if we scale back what you mean by "... or in a very large emailing service provider?"
Large is a relative term so let's say maybe not Gmail or Office 365. But something smaller, like a single company's exchange server. Many large corporations still host their own email in a single location and may or may not even maintain offsite backups. These... I've seen get lost forever; so, I can answer that.
The impact is normally felt on a per user basis. Some people do everything out of their email: They use it for keeping notes of who they are talking to, what they need to do, basically its their entire workflow and without it, they are a sunk ship. Other people only use it supplementally and would hardly notice it is gone. All of their schedules and notes may be in a CRM or a Spreadsheet or what not.
Even in a single company two people doing the same job will often have very different levels of reliance on it. So, on the small scale, some people will be really screwed. On a larger scale, the company would generally still continue to function since those who are less email reliant can generally be very useful in keeping things afloat until the other people are back in working condition. There will be some economic loose, but averaged across a large group of people, it's not the worst of things that can happen.
If you were to handwave away all the reasons you would not see all the email accounts in the world suddenly deleted, you'd see this same pattern on a much grander scale. Most people would find it to just be an inconvenience, some people would be screwed, but the mechanizations of our world as a whole would go on because most people would be fine enough with the loss.