That does not look like a problem.
One thing you might have to get used to is the polarity of meter probes on the resistance ranges on various meters. There is a battery in the meter, and it applies via the probes a voltage. But the polarity of that voltage that you are applying from those two test probes might be different for different meters for the red & black probes.
For example if you go to most analog meters the positive of the battery emerges at the Black probe. But in many digital meters it emerges at the Red probe, unfortunately there is no standard.
For any meter you own, you need to find out what polarity the voltage is on the probes on its resistance range, the reason is you might be using the meter on the Ohms range to test transistor junctions & diodes etc.
You can find out for your meter with a humble diode like a 1N4148 and a simple rule. The rule is "positive emerges at the line", meaning that say if you have a meter during a forward resistance test on a diode, and the meter design had a positive potential on the Red probe, then if that Red probe was connected to the side of the diode that did not have the line on it, and the black probe was connected to the side of the diode with the line on it, then conventional currrent (opposite to electron current) would flow from the red lead to the line on the diode where the black lead is, hence; positive emerges at the line, and you would see what looked like an Ohms reading, and when you reversed the probes, the diode would appear open circuit. But if the meter was an opposite polarity type, clearly the black and red leads would have to be around the other way to get the test diode to conduct. You can also use one meter on the volts range to detect the polarity of the probes of another meter on the Ohms range. Of course if a meter on the resistance range is applying a reverse poarity to a pcb's power supply terminals the result is even less meaningful.
In the situation you have in the photo, it is possible that the meter is supplying a tiny reverse voltage to the +5V rail ( because you do not know the polarity of your meter until you test it) but more than likely it has the positive potential on its red lead. The value you are geting is not at all unrealistic for a normal unpowered pcb with a number of IC's. It is not a pure Ohmic resistance, but it represents the load of many semicondutor junctions in IC's.
In any case for a +5V supply line, even if that remained at 306 Ohms, the power dissipation is only 5^2/306 = 80mW , so its perfectly safe to connect the 5V supply, what you are looking at is far from a dead short. Normaly Tanst caps, when they short, go to a low Ohmic resistance less than 10 Ohms and often less than 1 Ohm.
If you are worried about some pcb like this, one trick s you can do is power it from the 5V via a resistor in the range of 1 to 5 Ohms, them measure the voltage drop across that resistor and with V=IR and calculate the current, and see if that appears to be about normal, for the number and types of IC's involved.