Best answer: Synology DS1019+ can technically stream 4K media to compatible devices; just don't go and try to do some serious transcoding to multiple devices.
There's a slight difference between the decoding of media and transcoding it. The former is the simple process of the NAS streaming a media file to a compatible device like your PC or smartphone that supports the same file type. The CPU doesn't really have to do much work for this. Where things can get overwhelming for the CPU is if the recipient device (your TV, for instance) does not support the file type, requiring the CPU to convert it on the fly.
If you have a 4K TV or device on the same network as the DS1019+ and everything supports the same 4K file format (.mkv, for example), you're good to go and shouldn't hit any walls. The NAS will stream the media file to the TV or device and it'll play. If the TV or device doesn't support the file format, you'll need to transcode it. This NAS can transcode even 4K content, but you may encounter performance issues and it's not really recommended.
The Synology DS1019+ comes with an Intel Celeron J3455 processor. It's no slouch, packing four cores (no hyperthreading) and can boost up to 2.3Ghz, but it's simply not capable of handling the transcoding of 4K files. You likely won't find many NAS models available that can do 4K transcoding unless you have seriously deep pockets or the know-how to build your own NAS PC. The QNAP TVS-882 will get the job done in regards to 4K transcoding, for example, but it'll cost you over $1,500.
It's not a lost battle, however. The D1019+ can transcode up to 1080p for the devices that can't read the formats you happen to have stored on the NAS. Just make sure your output devices can handle the same file formats you plan on streaming and the DS1019+ will be a perfect Plex server.
One factor to keep in mind when it comes to hardware transcoding is you need Plex Pass to do it with the Plex Media Server. If you only have a free account, you'll need to rely on software transcoding. You may encounter some buffering or other related issues if relying on this method.
Paying out for Plex Pass will ensure you're taking advantage of the hardware you have available. You have the choice of three plans: $5 per month, $40 annually, or $150 for lifetime access. The one-off payment of $150 would save you a lot of money in the long run, especially if you have everything set up for years of use.
You can do everything and so much more
If you have a budget that could cover the expense of the QNAP TVS-882 and need a server that can handle larger 4K movies, look no further than this NAS. It has an Intel Core i3-6100 processor, but QNAP does have models available with i5 and i7 CPUs if you need even more performance.