Because of a broken power button on the front of my computer I've had to take the cables connected to the button and create a sort of "hotwire" setup to start my computer. This is done by simply pressing the ground and power switch cables against each other, and having those cables loosly taped to the outside of my case.
Could these cables contain any dangerous amount of electricity, if a person or cat would accidentally touch them?
The power switch provides an active low input to a logic gate within the power supply circuit.
This provides a low current logic low (ground) gate pulse to a BJT transistor biased to an isolated 5V source with the maximum current likely much less than 10mA. This portion of the circuit is also electrically isolated from the rest of the power supply circuit as can be seen in the diagram.
So to answer your question, No there are no dangerous voltages present.
Source: https://en.wikipedia.org/wiki/Power_supply_unit_(computer)
The wires should only have low voltages on them (<= 12V), so they're safe to touch.
However, you should be careful, because if one of the wires is connected to the standby voltage rail, shorting it to the chassis could potentially damage the motherboard.
The wires are safe to touch as they are powered from a 3.3V chip on the motherboard, and current is limited by a resistor.
However, if you carry an electrostatic charge, and you touch the wire, you will cause an electrostatic discharge (ESD). This is harmless for you, but it can zap the integrated circuits at the other end of the wire if they are not protected.
Normally, the manufacturer would have taken appropriate measures to ensure all input/outputs are protected against ESD, which begs the question: did they consider it worthy to protect this particular IO since it is normally not an externally accessible IO, but rather an internal wiring?
For example, looking at this motherboard schematic (pages 23-24), the power button signal is filtered by a 100nF cap which should be fine, however the RESET button signal goes straight to a IC pin without any protection.
While these chips include ESD protection diodes, such diodes are sized to protect against ESD events during fabrication, which are quite limited in charge as people manipulating chips are expected to take proper precautions. More robust protection diodes on every pin would use more silicon area and make the ICs more expensive, while providing no benefit unless the pin is actually accessible on the finished product and can be zapped by the user.
Therefore, your arrangement carries a risk of zapping the motherboard. It will probbaly be safe if the floor is tile and you are wearing cotton. However, if the floor is insulating, and you're wearing wool and sneakers, and the weather is dry, when touching metallic objects your fingertip can turn into a pretty impressive lightning gun!
Other answers have explained that the voltage difference between these two wires is almost certainly very low, and has next to no chance of harming you when you touch both of them - the more likely consequence is that you might harm your computer through static discharge.
There is another consideration for the electrical safety of low-voltage circuits though, and that is the voltage difference between these wires and ground (earth).
Your PC's power supply should either have its ground (0V) rail connected to mains ground (earth), or it should have adequate isolation between the mains side and the low voltage side making it a Class II device, and in either case it should be safe to touch the low voltage side even if your body is also touching something grounded. However if there's anything wrong, for example:
then the result could be a dangerous voltage, relative to ground, on the low-voltage side of the PC power supply. Just in case of one of these, it's safer not to touch the wires.
The logic voltage on your motherboard is probably ~12V or less, as said in @FakeName's answer; I'd guess ~5V in most cases. But, why guess?
If you're going to be messing with electronics, seems like you should have a multimeter that could directly test it. It'd seem safer to actually check the voltage than just assume.
As for the danger of electricity, in general:
A power source can assert a voltage differential (e.g., 5V).
How much electricity comes from that voltage differential depends on the effective resistance between the terminals.
The lower the resistance, the more current will flow due to the same voltage differential.
Most power sources are limited in how much current they deliver.
So, how much voltage/current is dangerous?
"How much voltage is “dangerous”?", SE.ElectricalEngineering
"Safe current limit for human contact?", SE.EletricalEngineering
To provide an arguably alarmist warning from @RussellMcMahon's answer (currently at +24 votes), while noting that this observation is extremely unlikely to apply to the current scenario:
FACT:
12 VDC CAN kill and has killed people.
While 12V is almost always safe, worst case situations can and have lead to death.
Mechanism may be ventricular fibrillation BUT paralysis of the respiratory muscles occurs at about 20% of the current needed to introduce fibrillation.
See discussion and references at the end of this answer.
@Neelix's answer above speculated that the equivalent power switch on a PSU may be current-limited to <10mA. 10mA of direct current would presumably be perceivable-but-not-painful, according to the table from @OliGlaser's answer here:
So, it's probably not an immediate hazard upon human contact in most reasonably foreseeable cases, but it's still probably something to measure and be careful about anyway.
As a person who has been electrocuted on many occasions by household mains power levels and knocked to the ground by a close proximity lightning strike I'll give this one a go.
18 mA directly across the heart can be lethal. That being said, there are technically enough mA stored in a small AAA battery to kill a person. However, it takes usually about 80+ volts to penetrate the skin of a human being. Humidity and salt content can decrease this voltage slightly. Licking a 9V battery gives a metallic tasting gentle buzz, but it is not painful. Holding the wires inside a electric-click cigarette lighter or candle lighter gives a small jolt (They produce upwards of 80 volts) with very low mA, surprising but not really painful.
Voltage determines if the electricity penetrates the skin. I've been hit with thousands of volts, not dead yet. Stun guns are 15,000 to 28,000 volts typically, but only a 2-5 mA. An electric fly swatter is about 5,000-8,000 volts, but again only a few mA. (Great DIY stun gun)
Amps determine the damage done. Amps that do not travel across the heart are almost never lethal. Electricity always takes the shortest route to ground (through your hand and down your side in most cases). I've taken 15,000 mA at least a dozen times, not dead yet, but it always very painful and alarming.
Frequency in the case of AC voltage determines the 'feel'. Around 60 Hz feels like electrocution, much higher frequencies feel more like a burning sensation (You don't feel like you are being electrocuted, it feels like heat or fire, and you are more than likely receiving nerve damage). Tesla coils are thousands of volts and Hz with very low mA, and are painless, but not completely harmless.
