Thanks for filing the issue!

Unable to reproduce the issue on reported chrome version 65.0.3325.146 and on the latest canary 67.0.3365.0 using Windows 7 and Windows 10 with the below mentioned steps.
1. Launched chrome
2. Navigated to https://mail.google.com/mail/
3. Entered credentials and clicked enter.
We are are to open the inbox of the mail with out any errors. Attaching the screen shot of the same.

@Reporter: Could you please have a look at the screen cast and let us know anything missed from our end. And please check the same in a new profile i.e., by creating a new person with out any apps and extensions. 

Deleted: 820297.PNG 121 KB

	820297.PNG 121 KB View Download

This is likely due to a buggy middle-box on your network blocking newer versions of TLS (TLS 1.3). Do you know if you have any anti-virus/proxy/firewall products installed that might be causing issues?

It would also be helpful to get a net-internals log when you reproduce the issue, see details here:
https://sites.google.com/a/chromium.org/dev/for-testers/providing-network-details

I have Avast Free Antivirus on my computer, but it recently expired. I also
have Windows Firewall, but Google Chrome is on the list of permitted
programs. I'm not sure how to fix the issue since all other browsers work
and Google Chrome works like normal on all other sites apart from Gmail.

I tried to complete a net-internals log. I've attached the file here. To
replicate the issue, all I did is open up Google Chrome and go to the Gmail
page. The error page pops up.

Does turning off Avast make any difference?

It looks like your attachment in comment #5 didn't upload successfully, could you try adding it again.

I will try the attachment again. After completing the net internals log, I
tried to open the saved file, but my computer said that Windows cannot open
the file. Is there something different about Json files?
I disabled Avast for 10 minutes and Google Chrome was able to access Gmail.
I could disable Avast permanently, but that has been my main free antivirus
software.

Could you attach it from the web instead of via email? Go to your bug report here:
https://bugs.chromium.org/p/chromium/issues/detail?id=820297

At the bottom of the page, hit the "Attach a file" link and select the JSON file. Thanks!

Additionally, do you know what version of Avast you have, and if you have any interesting options enabled or disabled? We tested agains Avast before and weren't able to reproduce any problems, but perhaps it's a different version or configuration.

Unfortunately, these sorts of antivirus tools are often rather buggy, and it's quite possible that Avast did not quite get things right here. TLS has very particular rules that implementations must follow to avoid breaking on updates, and our experience is these sorts of middleware products are often non-compliant.

Here's the attached JSON file. Hopefully it can be opened here.

Deleted: Chrome internals netlog dump.json 1.7 MB

Chrome internals netlog dump.json 1.7 MB View Download

I haven't yet found the version of Avast that my computer is using. I did
find that it was created and put onto my computer November of 2015. Does
that at least help us narrow it down?

Getting the exact version would be helpful since we're not able to reproduce it locally. Alternatively, if you can update or reinstall the newest version of Avast to see whether that fixes the issue, that would be helpful to know.

This article has some instructions on finding the version. Could you check there perhaps?
https://support.avast.com/en-au/article/34

Alright, I just downloaded the antivirus program again from the Avast
website, and the issue seems to be resolved. I'm able to access my Gmail
straight from Google Chrome. I'll try and keep tabs on it in case it
changes, but as of now it's working like normal. Thank you for all the help
throughout this process.

Best,
~Stephen~

Glad to hear its working for you now, and thanks for verifying that the Avast update fixed it.