You can survive on something like oil for an emergency (as in a very short time) so long as you define “survive” as “slow the process of starvation.” Your body will soon become depleted of various nutrients and minerals if you’re only consuming oil.
In an emergency, though, the priority is water. Depending on circumstances (weather, physical effort, etc.) and one’s health, a person can survive without food for a couple of weeks to a couple of months, but can only go without water for a couple of days to about a week without water (in extreme cases, two weeks without).
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