This is not generally possible without additional mods, as the temporary files are not really stored anywhere readily accessible.
However, with client-side mods it is possible to download the loaded chunks into a world file for later viewing offline. I have used World Downloader in the past to rescue some of my larger builds from a server I no longer play on for my single-player enjoyment. Once you begin 'downloading' it will save all the currently loaded chunks, and any further chunks that load as you move (if you travel to the Nether or End, it will download those too) until you tell it to stop.
There are a few caveats: some red-stone contraptions may need to be manually re-set, and it will NOT download the contents of containers (chests, hoppers, dispencers, etc) unless you open the container while downloading. Also, all the coordinates of everything downloaded is preserved, so if you only downloaded the chunks around X=10,000,Z=10,000, that is the coordinates they will be at in the world the mods create. I believe it records the position you were at when you finished recording and will set your location to there for the downloaded world, but if you don't find a bed or otherwise set your spawn-point it may send you to the default spawn location of the server if you die (which may be a ways away from the downloaded chunks).