As described on the minehut website under How do I join a server (https://minehut.com/go/howto-java), You can connect to the minehut lobby by adding mc.minehut.com as a server. This is where you can execute your commands etc. Once you have connected to the lobby, you can execute the commands from within minecraft. Get Started Open Minecraft and click “Multiplayer”, then click “Add Server”. You’ll be taken to this screen, where you’ll enter “mc.minehut.com” into the “Server Address” field. Give it a name (such as “Minehut Lobby”), then click “Done.” Almost There… The Minehut Lobby will be added to your server list for easy access. Once it’s connected, select the server and click “Join Server”. That’s It! You will be automatically transported to the Minehut Lobby, where you can chat with whoever’s online and log into your server. Have fun!

There exist tools called "world downloader". As you explore the world these will save the chunks locally so they can be loaded into a new world. It seems there are variants both as client mods and as seperate tools that sit between client and server. There are some caveats though, first that areas you do not explore will not be saved. Secondly I believe chest contents are only stored on the server side so they can't be saved unless you open them. The other option would be to ask the servers admin nicely if they will provide all or part of the data files. You will also probably want the world seed so that newly generated chunks line up with your downloaded chunks, again you can either ask the admin nicely or there are tools to reverse engineer the seed. Disclaimer: I have not used these tools myself and cannot vouch for thier safety or which version/fork of them are best (hence why I have linked the google search instead of any particular fork) it may also piss off the admin of the server.