Here's what I need:

Three websites, in separate subdirectories

example.com/site1
example.com/site2
example.com/site3

- With a template shared between all 3, so if I update the template on one the others will reflect update also.
- A shared images/media section
- The sites will all have similar content but with significant text changes.

I'm hosting on a linux apache server.

Is this possible with JMS multisites? If not, is something close to this possible?

Thanks!

Yes you can do that with JMS.

JMS is able to recognize domain, subdomain and subdirectories.
So you can have website like
domain1.com
domain2.com
subdomain1.domain.com
subdomain2.domain.com
domain.com/site1
domain.com/site2

All is possible.

By default, JMS share all the files and folders between the websites (when the symbolic link is allowed).
So that you have the same /images and /templates directory.

So yes all that is possible with JMS.