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Apperently more like a plugin. Pure web technologies don't give you more than a couple 100mb of local storage (at best).

I don't know whether Superhuman will be a plugin or not, but there's no reason it would have to be; IndexedDB can provide gigabytes of space, and it's standardized, and current versions of Chrome, FF, and Safari all support it.

Depends if you consider IE and Mobile Safari as a required target. They have 250MB and 50MB limits, respectively. But yes, if these can be omitted then IndexedDB API seems fine nowadays.

Edit: MS Edge also has 250MB-500MB; they're just always playing catch up, even when their market share has dwindled...

Does the spec define the minimum amount of space the browser must allocate?

Can normal webpages access IMAP and SMTP servers?

No.

Like gmail.com or outlook.com, if they were limited enough to support just one browser..

No, that is not a website or webapp, it's a Chrome app.