It is not possible to play copies of games on a legit, unmodified 360 - they will just not be recognized by the system. Of course, you can always modchip your console, but this is not free, is considered illegal, voids your warranty and may result in Xbox Live ban.
However, I would not worry too much - original DVDs are pretty sturdy and don't break easily, also the 360 DVD drive isn't very vulnerable, unless heavily used to play copies of games (burned DVDs wear the drive significantly faster). There is one exception, however - never move a console while turned on/playing. The 360's DVD drive is known to scratch DVDs when moved during reading the data from disk.
In your case though I would recommend buying an official, standalone 250 GB Xbox 360 S hard disk drive and installing games on it. Not only does it allow for faster loading times, but it also reduces stress put on disks and drive. Also buying games on Live and launching straight from HDD would surely resolve disk problems.