Oranges can be kept at room temperature. The start to decline in quality within a week. They will still be good until mold pierces their skin and spoils the inside. This can happen quicker if there is moisture present. Some fancy mail order companies wrap the orange in wax paper to keep out moisture. This can help some. I am not aware of any tricks to prolong storage time.
You can store oranges in the refrigerator crisper for 2-4 weeks. Again the quality will be good until mold grows.
Orange peel, juice or flesh can be frozen. Frozen citrus will last for 6-12 months.
Oranges cannot be successfully canned in any way, except marmalade. It becomes bitter when canned.