You can pay Iron Mountain to do it for you, take the tape to a safe-deposit box, or take it home. Most important items first: make sure you take backups offsite! I know that's not mentioned in your question specifically, but the reminder never hurts. It really depends on what you're backing up and how important it is to get that data quickly. If it takes more than one tape for a full, you'd adjust as necessary, and obviously you'd need more tapes up-front.
You'll need more tapes w/in a month if you want to do something like this.
BACKUP EXEC 20 FULL
When we were only using tape, we'd keep the last full of each month for one year, and we'd keep two of the fulls forever. I'd keep at least the last 4 fulls, and I'd have at least 3 or 4 diff tapes to rotate through. You say you have 5 cartridges, but LTO3's are cheap enough now that you can buy some more. If your full backup fits on one tape and all the differentials also fit on one then you'd get something like:
I'd have separate tapes (or sets of tapes) that I use for full backups and for the differentials. Tape rotation will depend on how much data.