First, you just need to narrow down whether the NAS will support Plex at all. These two pages might help with that:https://support.plex.tv/hc/en-us/articles/201373823-NAS-Devices-and-Limitationshttps://docs.google.com/spreadsheets/d/1MfYoJkiwSqCXg8cm5-Ac4oOLPRtCkgUxU0jdj3tmMPc/edit#gid=314388488
Note: The second link is a Google Docs page directly linked to from this Plex page: https://support.plex.tv/hc/en-us/articles/201373803
From the compatibility list, it looks like you are going to eliminate all the cheaper NAS options as they don't support even transcoding video to even a single client at all or at best to 480p...so they will definitely not handle multiple streams. For example, the ReadyNAS 316 that I have that is a small business class NAS that costs around $600 with no drives can only kind of handle 720p to a single client. So, you are likely looking at an NAS with an Intel processor in it, which is almost definitely NOT going to be on the cheap side.
So, you might find that you are better off just buying a relatively cheap computer that you use as a dedicated Plex server (after all, NAS devices are effectively just cheap computers that typically run a form of Linux). And if you wanted NAS functionality, then you could just run FreeNAS on that computer...from the Wikipedia page, it does look like it supports a Plex plug-in.
You can also take a look at this post on the Plex forums to see if it helps:https://forums.plex.tv/discussion/113690/most-affordable-nas-with-1080p-transcoding
You also might either further search the Plex forums or even post your query there.