Which part of your email system is being targeted? What specifically are they targeting? Are they just making connections to a port, are they sending enormous amounts of spam, are they trying to brute a password?
Inbound mail traffic should be routed through lightweight MX servers. It's a common mistake to put an MX and a mail store on the same system. You can deploy new MX endpoints all day long and just update your DNS for it and email will still work pretty well.
Although there are perfectly reasonable arguments against it, Fail2Ban or similar can shut down nuisance traffic on a mail store. You should beware though that it's difficult to ensure that Fail2Ban or other active-response log monitoring can't itself be abused to ruin your day.
If your mail store is just getting hammered, it's a tricky problem to solve without a lot of resources. If you're the only user on your system, there's no reason to publish a dns record for your mail store, so move it to a new ip and update your settings. If you can't do that and Fail2Ban can't resolve it, you get to start thinking about things like distributed mail storage. I've been wanting to check out dbmail for years (https://github.com/dbmail/dbmail/), maybe you can give that a whirl and link a writeup of your experiences with it.
I have experimented a bit here and there with dovecot-on-mysql and multi-master percona and all that and it's not fun or reliable.
If you're getting mind-blowing amounts of spam -- especially if it's newsletter signups -- it's possible you're being mailbombed. That sucks, there aren't a lot of good solutions for that right now, even Gmail users can be victimized by it. It seems to often be associated with some financial fraud, probably because those suspicious activity notifications kinda disappear when you're getting 20,000+ messages a day. I'm working on some software for this, it's in limited testing now but still really rough. Email is hard to write good software for.