How long can I use a computer without a fan

You need to replace the fan as quickly as possible. Without the fan, heat will build up and fry the electronic chips.

To be honest, as long as it takes to burn out the processor

If it's a case fan then you will be OK for a while mainly depending on what in there, but normally you can just open the side up and make sure there is good air flow. If its on the; CPU, Chipset or Video card. Then I would give it on average about 1.5 minutes to do irreversible damage. And last, if it's the power supply, I would give it maybe an hour, again depending on your usage. But a bad power supply fan will usually only hurt the power supply, but you usually have to replace the whole supply it if the fan goes bad, unless you car conformable opening your power supply and hoping the fan is not soldered onto the board