How close can I make different monster spawners to be able to work

Mob spawners activate when within 16 blocks (sphere) from the player. Any number of them within that range will work, with the obvious caveat that the spawner itself is not a spawnable space - the mobs need empty volume to spawn. The more spawners you put within the 9x9x3 volume centered on a spawner, the less empty space for mobs to spawn within the volume you have left.

But finding more than two spawners within 31 blocks from each other (so there's a location between them where the player activates both) is quite rare, so it's not a big worry.

On the other hand, if you plan a general mob farm, you'll be better off with a natural spawning based mob grinder than the spawners. Spawners only spawn up to 4 mobs every 25 seconds on the average (between 10s and 40s random cooldown) and only of the type the spawner is, while a good general mob farm will spawn all, or at least most kinds of mobs and its speed is limited only by your ability to kill them fast enough to keep the number below mob cap.