As Caja stands right now, no. If you really want it, I might look into implementing it for you, since I now have pretty good ties with the Caja community.
Would be nice if you could. Maybe users that have a right-to-left language may benefit with this too by just setting up a right sidepanel.
Already, for right-to-left users the sidebar appears on the right side. I know this because I tweak around with GTK (the graphical software library underlying Caja) a lot, and I've switched to right-to-left mode in the past. In GTK, stuff is not expressed as "to the left" of something else; it is expressed in terms of numeric positions. So in all cases the sidebar has position 0 and the main content of the Caja window has position 1. In left-to-right languages, stuff with lower position numbers goes on the left side; in right-to-left mode, stuff with lower numbers goes on the right side.
The difficulty is that, for decades now, GTK has had a widget (user interface element) for a two-paned container, but not a three-paned container or the like. If you've ever looked carefully at applications that have left and right sidebars, they actually have a sidebar contained in a "paned container widget", then another paned container widget which contains the main content and another sidebar to the other side of the main content.
I would go with the conventional approach, but I can easily see someone asking shortly after I implement this: "Hey @gordon: I have notes on files and I would like to view the notes horizontally, since I write long lines of notes. Would it be possible to add sidebars to the top and bottom of the Caja window instead of just the sides?" At that point we'd have 4 paned container widgets all nested one inside another, and then someone comes along and asks for two sidebars both on the left side, and then someone asks for three sidebars on the bottom and three to the right...
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Ah didn't know that about the sidebar positioning. Good that its already done for RTL users.
And yeah, I can see caja 1.30 now, sidebar for the sidebars haha. 
Well, here I am again giving an update on the secondary sidebar feature request.
This update is mainly for the benefit of @Cold, who originally requested the
feature.
As usual, my personal life has gotten in the way of programming the feature
addition for the entire working week. However, now I have a better idea of
what needs to be done to implement such a feature. I think at least initially
I'm just going to give you a patch you can apply to Caja (like what I gave to
@Guy above) which will enable a "secondary side pane" which appears on the right
of the Caja window (or the left, if your preferred language is written right-to-
left). I was going to give you a patch which would allow you to add sidebars
anywhere and everywhere, but I don't want this to drag on forever.
Speaking of right-to-left languages, I have proof that the layout of the user
interface of Caja changes if your language is written from right-to-left:
The screengrab also proves that some themes contain minor bugs which display
elements of the user interface wonky if the application is in right-to-left
mode: This is the BlueMenta theme, and you may notice that in the Caja pathbar
(just below the toolbar), the "scroll buttons" (the buttons with the black
arrows) at each end are shaped incorrectly; ideally, they should adapt to a
right-to-left user interface. This adaptation is possible, and I plan on
starting yet another one of my famous issue reports / pull requests in hopes
that this issue can be resolved. Following is another screengrab, of the same
Caja window but this time with my theme fixes applied:
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That's great @gordon, thanks! I am left-to-right, but brought up RTL readers in case they could benefit too(though, now I know they had it already).
Also, don't worry about taking long time to getting around to doing it. Personal life, and work is priority . 
Well, it's quite unfortunate really, but I need to tell you the truth, @Cold:
Adding a second sidebar is going to be a real challenge because the code was only designed to handle one sidebar. I would go through the contortions required but I'm not getting a break from everything else -- I just don't have the time or the money right now to do this.
Therefore I'd suggest starting a Caja issue requesting this feature. @franksmcb gave a link to the Caja issue page up above.
If you have no GitHub account or something, I can start the issue for you if you want. But I just cannot continue on this particular endeavor, short of being monetarily compensated for my work.
Sorry for the late reply, work got in the way.
Dont worry about not doing it. It was more for having to fiddle less when checking stuff in caja and also a way for RTL users to have a better time with a file explorer. Since you told me they already have it, the need for it went way down.
Still, thanks for giving it a look.
I'd hate to sound like a self-promoter, but this is mostly for @Guy's benefit:
I'm not sure if you've noticed, and I'm not sure if you use Ubuntu "intermediate releases", but my Bookmarks sidebar patch (among many other things) was officially integrated into MATE 1.26 in August (actually, technically, into MATE 1.24.2 around January, but nobody noticed it then), and MATE 1.26 is officially in Ubuntu MATE 21.10; so this feature request is now an official part of the MATE Desktop, is in at least two stable releases of the desktop now, and is officially available in Ubuntu today.
In other words, if you upgrade to 21.10 (or 22.04 in a few months), you'll no longer have to compile my patch yourself.
Just thought I'd let you know. Best wishes.
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Small note.
For 20.04 LTS it is possible to get the patched Caja (and all full new MATE 1.26) using fresh MATE PPA as follows:
sudo add-apt-repository ppa:ubuntu-mate-dev/fresh-mate
sudo apt update
sudo apt upgrade
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