Did you buy a 360 with only the 4gb flash storage built into it? If you did, that is why you cant save or view replays. You will need to buy a harddrive for whichever version of the 360 your purchased.
http://www.amazon.co.uk/Official-Xbox-500GB-Replacement-Drive/dp/B00MOLZ88W/ref=sr_1_3?ie=UTF8&qid=1420214275&sr=8-3&keywords=360+xbox+hard+drive
This one is for the 360s and 360e models.
Only buy genuine microsoft official harddrives.