Is it possible to deploy corporate mail in the cloud?

Question

Mail server

Good day everyone!
We are currently using postfix. IMAP+SMTP only (no POP3). Email client - Thunderbird. Installed on a terminal server, configured using autoconfig - the setting is quite strict - users can change the minimum - only the most necessary. On the server (in the profile of each user) some cache of letters is stored (I don’t know how to disable it or it’s impossible), but the letters themselves are on the mail server.
What does not suit:
1. Not reliable. If "tomorrow" our mail server falls, then there will be "trouble" in the company.
2. There are no server individual address books. General - yes.
What do you need from the new "mail server". In quotation marks, because I still don’t know what it will be ... maybe Windows Azure in general.
0. Reliability at all levels.
1. Authorization of users only in the domain.
2. General and individual address books - on the server.
3. Access outside the office - only to the elite.
4. IMAP.
5. Users up to 100. The volume of each box is up to 60 GB. The number of letters in one box is up to 200,000. Yes, we keep correspondence with counterparties for several years ... nothing can be done about it.
6. General box - "CHECK" for collecting the archive of all mail (incoming and outgoing). It is very necessary in case when the company is fined by the client due to non-fulfillment of the order, and the manager Masha says that she did not receive the letter, because the IT department - shit - did not ensure the uninterrupted operation of the mail.
7. Filters (sorting) by the server (or client). Now only the client sorts. Not a very critical point.
8. Aliases, of course.
Perhaps I have described our tasks too roughly. But I described them based on what is missing now.
I suspect that no corporate mail Mail.ru (Yandex Google) will suit us due to the lack of a full-fledged implementation of everything that I described. Probably, you will have to look towards some kind of cloud (or two or three different clouds) in which Exchange should be deployed.
Please tell me what is on the market now, what is worth considering in more detail, and what is not worth looking at at all.
Thanks for the help.

Answer 1

1) Backups will save the world and your company.
2) Microsoft Exchange for rent is given by a large number of providers, just like in Azure, switch to Outlook and you will be happy.

Answer 2

I slightly corrected: "The volume of each box is up to 60 GB", but it was "The volume of boxes is up to 60 GB".

Answer 3

Zimbra OSS + Zextras Suite. For your needs - that's it. Perfect solution. The virtual machine provides proper backup, and Zextras Suite backs up to another server. Moreover, if you have created an AD or LDAP database, you can set up autoprovisioning.