Update (Oct 5, 2021): Never mind. The advice below might still be helpful but the problem eventually returned, and then finally went away; I suspect now that it may have had little to do with the volume of chat content (though it may have exacerbated the problem) and (almost) everything to do with an issue by the Skype service itself.

I prefer Skype for Web in a browser window over the desktop version because it is less intrusive on incoming calls. (Old Skype popped up a discreet little notification in the lower right corner, but Skype 8 steals the desktop focus, which is ultra annoying.)

However, lately Skype for Web developed this annoying habit of crashing repeatedly, with the message

Oops, it looks like something went wrong.

Click OK to automatically reload Skype for Web.

Needless to say, restarting Skype in the middle of a conversation is not a very helpful thing to do. Worse yet, even if you don't click OK, simply switching to another browser window causes the Skype for Web page to reload, dropping the call.

I checked and the problem persisted across browsers and computers. Yet when I logged on with another Skype account, the issue disappeared. This led me to suspect that the real culprit might have been one of my extensive chat conversations, with a multitude of messages and several rather large (file, animated GIF, video) attachments.

So I cleared the history of the suspect chats and indeed, Skype for Web has been running reliably since.

This, then, is the solution: if Skype for Web crashes in this manner, and crashes regardless of which browser you use, check if any of your conversations had excessive amounts of data.

NB: You can download cached conversations and files from Microsoft.