Of course, that isn't persistent across tabs, but it's a neat thing to try out nevertheless.
Native support for a sidebar was declined, possibly delegating that to extensions. The functionality enabling extensions to do this hasn't been implemented, and at one point looked like it never would, but a proposal has been under development (preliminary documentation here).
It's been suggested that the dev team is instead focusing on the panels feature (video demo), but it hasn't landed on the browser yet. There's the Panelize extension that promises to allow any page to be loaded in a panel, but as far as I know it only works in ChromeOS.
TL;DR: No, it's not possible yet, but some workarounds may do the job depending on your needs; and support for a proper sidebar (through extensions) is under development.