As everybody says, there's unfortunately a couple different problems with accessing modern Internet services on vintage Macs. You either need to alter the service, use external help, or use/run services that are themselves, vintage friendly.
The first set of problems really comes down to Encryption
generally not being "optional" at all these days. 68k Macs have a difficult time handling the flow of data in an "everything is encrypted" setting. Especially with modern encryption standards. They usually
can do it but it is very slow.
The other set of problems comes in play when you start talking about offering services optimized for vintage computers. (This really generally applies to any computer from before around 2005 but due to certificates changing you can also see this stuff in software that hasn't gotten security updates in a few years.)
Old computers expect implicitly for everything to be trusted and for authentication to be optional and rare. Most old email clients don't have a way to ask for SMTP auth, and most old email servers don't require it and don't have a way to enable it.
In addition, most modern email servers do not trust servers that don't have certificates.
The third thing (I know I said two above) is that most modern email accounts are a veritable firehose compared to the IV drip of the way email worked in the '80s and '90s, or even the early '00s. My personal and work email accounts each get about 100 total messages per day. It would be more if I explicitly asked services and web sites for email notifications. My work email account takes about 7 gigs on disk and my newer personal one (the one I use with almost no automated services) is about ~5-8 or so total gigs.
It's outright unreasonable to attempt managing 5 gigs of email in one go on a Classic Mac OS computer. Almost any email program on any release of Classic Mac OS will just absolutely lose it with that much email. Heck, Macs didn't even support 4-gig volumes unless they have an '040 or better.
The fourth thing (yes, I know) is that most modern email has lots of imagery and styling that existing programs can't display and that would in general look bad on small displays. (Again: if the HTML engine in your email client were capable of rendering it.)
It is worth considering that the setup you are asking for is a security nightmare and there is no practical way to do it without opening yourself up to a lot of vulnerabilities.
Fun anecdote. I run a Mac OS 9 web and file server --
http://vtools.68kmla.org. I was originally planning on adding email to it.
As part of testing for that, I set up DNS and reverse DNS and got my ISP to open port 25, etc etc, turned the service on, and in under 24 hours had a bit over 4 gigs of backlogged mail traffic.
It turns out that basically every IP is always being scanned and AppleShare IP 6's email server software doesn't have any way to authenticate users, so it's just... open.
For better or worse, the Internet was radically different in 1998 to the wya it is now, both in severity and scale. The reason I noticed was because AppleShare IP's mail stack crashes (badly) if the one and only data file grows to more than 4 gigs. (Fortunately, at some point I'd set it to hold outgoing mail, so my IPs weren't, as far as I know, blacklisted.)
The other thing I had to do was ask other people I was testing with to allow the server in on their servers, which you can't guarantee public providers (Google/Microsoft/Apple/Yahoo et al) will be willing to do in perpetuity.
There's some work-arounds, you could set up a smarthost so that the email server looks modern to the rest of the world. And, you can put vintage access behind a VPN -- even an insecure VPN such as an unencrypted PPTP host will protect the mail access ports (in that case you're doing VPN to protect network ports, not to actually create privacy.)
Unfortunately, I haven't been able to find a PPTP client for any older than Mac OS 7.5.x on an '030 with 8 megs of RAM or better. You could use an older router to put an entire gaggle of vintage machines on a VPN, but I haven't tested or set that up and my home Internet isn't in a place where it would be reasonable to do that.
The next best option, I'd say, is to set up your email in a modern linux console client like alpine or mutt and use
ssheven on a 68k Mac to connect to that machine.
I wonder if mail could work as a sort of retro-challenge-ish bodge:
I have done this and I can not emphatically enough say that I recommend against doing it. You'll pay 2x as much for business internet, your IP will get scoped instantly and you'll instantly receive literally more mail than ASIP can conceive of, and then your IP will be blacklisted.
This
may be possible if you use a newer server OS but in general, I recommend against doing this unless you are an experienced email administrator who understands how things flow.
There is one thing left that still works and that is IRC.
We have an IRC channel for anyone who wants to hop in, vintage or modern: #68kMLA on irc.oshaberi.ne.jp.