Browser based access is absolutely possible. Not only that it exists.
It has 2 forms:
If you want a read-only version of the network there are a couple servers that do that. It's useful for showing people who aren't on ssb interesting conversations.
If you want to actually interact with the network that's possible too but there are... limitations / issues.
There are 2 aspects to this.
1) it's a distributed network. So, either you need to stick a client in the cloud with a web interface (Patchfoo [1] is just such a client, but the interface is.... meh) OR you need to run the entire client in your browser, and the problem with that is where do you store the data? You're going to quickly exceed the max of local store (i forget what that is but if you start following lots of people and they post photos you're going to exceed it).
2) You can't afford to loose your private key. SSB uses Public Key Encryption and as such if you loose your key you can no longer prove that you're you so you become disconnected from all your prior posts and people following the old you no longer get updates from "you" because "you" can't post as the old you anymore.
If you use multiple browsers then how would you sync the keys between them?
Also, SSB uses a blockchain for each user. If you have the same keys on multiple computers and one computer posts before it's synced up with the last post from the other computer then you've just forked your chain and it screws up everyone who's following you because you don't know which version of the chain each follower has ended up on.
[1]: https://github.com/ssbc/patchfoo