Disable mob spawn in a specific area of a multiplayer map

This doesn't disable spawning, exactly, but it will kill all mobs within a certain radius. First, you have to create a team for your friends and you to stop yourselves getting killed when you enter that location. So, run a repeating command block with this command:

/kill @e[type=!Player,r=RADIUS]

Replace RADIUS with the amount of blocks you want the command block to check for in each direction. For example, if you had a 5x5x5 room, and the command block was in the center of the room, your radius would be two.

So, now, whenever mobs that are not players spawn in that area, it'll kill them.

However, mobs count as both hostile ones (like creepers) and docile ones (like chickens). If you, for example, want sheep to survive in that area, change the command to this:

/kill @e[type=!Player,type=!Sheep,r=RADIUS]

So now, the command block will allow sheep to spawn in that area. If you wanted to add a pig, do the same for the entity type Pig:

/kill @e[type=!Player,type=!Sheep,type=!Pig,r=RADIUS]

And so on.

If the command block output in the chat is getting annoying, type this into the chat, not the command block:

/gamerule commandBlockOutput false

Hope this helps!