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It should already have a Import/Export functionality when going to the extension's Settings/Options page, clicking "Reveal advanced options", and using the self-explanatory buttons there. It should output JSON, which is also useful and can be manipulated, for instance, to compress the page tree contents to just unstructured plain text, possibly by using command line programs such as jq or jsontool.

You may see this explained here: Sidewise is great - looking for a way to backup & transfer

Beware of some bugs, such as these:

If Sidewise is not appearing, maybe try clicking the light bulb icon in Chromes tab bar. I do this sometimes to 'refresh' the data Sidewise contains, but be careful with doing that. I try to export before 'refreshing' because it sometimes loses recent page data that way.

Actually, it simply prevents my from exporting at all, due to a JavaScript error. I can try to post it later., but I think it has to do with a localStorage limitation like you said.
https://developer.chrome.com/apps/offline_storage - Local limited to 5MB? I thought I remembered a number like 400,000 (Kbs/Bs). The filesystem API seems to allow it the 'query' for more storage, if needed, but Sidewise does not support this.

I wish it would at least allow you to copy the backup data that includes what it saves to localStorage so we can import it later.

I also still have this issue on Mac OS X El Capitan 10.11.3, with Chrome 49, and other OSes.

It's been happening for a while, and it seems to be related to a tab de-sync issue. This may be caused by Sidewise not staying up-to-date with Chrome, and the browser may have changed or add tab events that Sidewise does not handle.

This may appear after some actions such as normally creating tabs after a while, or though ones created automatically by extensions, JavaScript, pop-ups, or otherwise.
This may be seen as the tab does not appear in Sidewise, but can be seen in Chrome's tab list at the top. Navigating to it does not highlight any row in Sidewise, and cannot be searched for.

A better culprit however, may be Chrome's Recovery or Restore feature, trying to re-open all tabs that may have been lost due to a previous crash. I'm not sure, but this may have been introduced at a certain point after Sidewise started development, and just wasn't accounted for? Just speculating...

I may have even noticed this on some other operating systems, and tried to prevent it from automatically re-opening these tabs. If I was successful in preventing that, I could see the old tabs in a hibernated window in Sidewise. This would be useful, and to keep Sidewise in sync with my browser, and to continue using the integration, I could simply wake the window and use my old session from there.

I have also found a way to restore single pages at a time, but it is a bit time-consuming, and not effective enough for dealing with many pages at once.
To do this, I simply open a new window, drag the target tab to there, close the tab, and the reopen it to have it appear in Sidewise. Then I can move it back to the original window I was working with to some varied success.

I would rather like to see this issue fixed, but I see development has stalled somewhat, and we may just have to either fix it ourselves, or use an alternative such as Tabs Outliner.