We tried that; it was called IRC. As far as I can tell, Slack is still little more than a hosted IRC service with its own web and mobile clients. Virtually none of the functionality they provide couldn't be replicated this way; it's more a matter of branding than anything else that they eschew the label "IRC".
Slack even provides an IRC gateway so you can access it using IRC clients. I don't know if their web and mobile clients are using the IRC protocol, but from what I can tell, they might as well be, which is the whole point.
IRC does not have this. I've to connect to your server, or you to mine. This doesn't scale as an email/IM replacement; it only works for rooms with a certain usage/target.
The huge problem with any replacement (aside from federation) is the network effect. No matter how good your solution is, you need critical mass for it to pick up, and no huge company is going to invest into getting users into a federated network which they won't control. There's no profit for the company.
IRC most certainly does have this. All larger IRC networks are exactly that: Networks of federated servers. It's been like that since the early 90's, when it was an absolute necessity as none of the ircds could handle all that many individual users.
Indeed, it’s the main catch. If at certain point ago critical fraction of people (‘everyone’) all were using X for what we currently use email for, be X IRC or anything else, then X’d probably be the today’s email.
Services such as the aforementioned Slack seem to be going maybe the only pragmatic ‘email disruption’ route. In corporate world you can have the central authority say “everyone use this or you’re fired!” and thus bypass the network effect barrier.
And as always, there is a huge gap between "X is just A+B+C" and "Here is an actual X". Slack may have not done anything on the transport side, but they did make the client better. AFAIK the only IRC client with full text search on the whole history, with support for files and images, is irccloud.com. That's not a lot. I wouldn't say IRC does everything Slack does.
IRC ain't dead yet. For example, there are IRC channels for many of the projects within Mozilla: