Wow. This is pretty much impossible to answer because there are so many factors. The only safe answer is, "It depends..."
1) It depends what type of backups you are planning to do. You are going to need at least a full copy of the database, which is 20 terabytes, but what sort of recovery window do you want? How long ago do you want to be able to recover too? If you need to recover to any time in the last month, then you are going to need a lot more space than just recovering to the last know good state.
2) Data Volatility: How much of your data is changing. If it is just a few records a day, then you will approach the backups differently to large percentages of change per day.
3) Downtime: If you are not allowed to have any downtime, your options are different compared to a "load once a week and do cold backup" approach.
4) Backup Location: Disk backups will be quicker, but may be very expensive for large databases. Tapes may be cheaper but slower. Assorted enterprise backup solutions are available. You probably need to speak to some vendors about available solutions and costing.
5) RAM: Allocate approximately 70-80% of your physical RAM to your Oracle instance (SGA+PGA). From there, monitor and adjust. This of course assumes that the only thing running on the machine is Oracle. How much RAM to use? Depends entirely on workload and required performance. Impossible for me to predict.