It’s not currently possible. The way I see it is that it would be used as CDN only for the larger content. So video, audio, maybe large photos. I don’t think Storj will be efficient for hosting an entire site on it as it simply works a lot better/faster with larger files and latency matters less.
CDN like functionality is discussed in the whitepaper though and with some tweaks to RS implementation and dynamically increasing RS settings based on demand, it is definitely possible. You’re right about it needing the browser as an uplink (unless it’s used in an app I guess). JT wrote a little bit about that here.
It probably won’t happen very soon, but at least they’re thinking about what is needed to get there.