Oil wells never run dry, so you shouldn't get completely stuck. You should prioritize building roboports until you reach a good oil field, don't spend too much blue science early on. I usually rush for advanced oil processing (and laser turrets), then I pause science for a while and focus on roboports for expansion.
Oh, maybe you know of this already, but using burner inserters to feed boilers greatly help on blackouts. You can also create a buffer with fuel (using burner inserters to both put into and take out of the chest(s)). I even use a requester chest for this buffer, set it to request wood which is then fed to the boilers. If you run out of fuel then you of course have a problem regardless, but you can avoid the problem of the electric inserters running out of power, effectively ending your power production.
In the long run solar is the safer power production alternative, though.
