Firstly unless you have a lot of duplication in that 200TB you can't.
Two areas you need to ask yourself:
1) What is the daily amount of change per server and with that the total amount that you need to backup daily or whatever frequency you need).
2) What is the initial uniquiness in the data being backed up. Mostly core OS and system with standard same local copies of data tables?
With that you can work out what you are trying to do form a need perspective. May be case of local imaging servers and doing differential backups automated. But if compression and duplication are not the case and/or high frequency of data then you will not be able to fit that data into a smaller area. You may only get 2:1 or less compression it is realy data centric and you need at least 4:1 and actualy better on all that data.
Look into the problem and define it better, with that a solution will be easier and may be case of proving that you can not do what you need may even need 400TB to back up those 200TB servers due to high volume data changes and need to be able to go back more than the last backup copy and may need to go back a month. Define the buisness needs, impact and with that the cost and budget as this is very much a case of working to a limitation when that limitation is not defined and may well be articifical and defeating in the objective to a cost that exceeds just adding more storage.
In short, define the problem better with regards to data and busness needs for backup -- hourly/realtime daily weekly and the type of data and how dynamic that data is. May be case of easy to do as realy only 1TB of data that changes across all servers each week, just do not know.
Then do not go about doing yoru own backup solution from scratch, as something to replace what you have worth a look if you realy know what you are doing and you may not. But do not reinvent wheels, many options/solution out there but until the true problem is defined then no easly solution can be given.
Heck do not even know if these are Windows, Linux, AIX, BSD or a combination of different servers and with that the options change. ALso factor in offsite aspect of backup as one electrical zapping from a storm or a fire could mess things up if that backupserver is sat next too the servers it is backing up. So many things to consider but above all sit down and define the problem in more detail as sadly the devil is always in the detail and better definition of the problem allows for easier solutions and over all less work as well as smarter work. It is a good leason too learn, just don't do it the hard way and write a backup system backed system on a fag packet problem and design.