The 4.2 v is totally safe. It cannot push anything close to a dangerous current through the human body or any part of it. The only (unlikely but possible) danger lies in the (old!!) tranformer developing a short between primary and secondary which could connect the 120v through to the 4.2 v side. The 120 v side is probably grounded to the case of the charger and that to the supply ground (earth), so if you also make sure that one of the 4.2 v terminals is also connected to the case of the charger then all will be well.

1 amp = 1000 milliamps

600 milliamps can stop a persons heart. In other words kill them.

When working with electricity the best practice is to consider any amount of electricity potentially dangerous.

You should always fully disconnect power to any electrical device you work on. Some devices have capacitors which can hold an electric charge even when they are unplugged. Many of these capacitors store a large enough amount of electric charge to kill a person if they accidentally touch it. Many of these kind of capacitors are in old Tube screen TV sets and monitors.

If you want a device that is relatively safe if you should accidentally touch it, the miliamps should be below 400. This can still hurt you, but it greatly reduces the possibility of fatal electrocution.

Voltage is far less important. The volts are what you might think of as electrical pressure. Lets say something had 100 volts and only 300 miliamps. This would give you a very unpleasant shock if you touched it, but the possibility of serious injury is very low.

Touch what?

If the input is concealed and out of reach, then it is quite safe to touch the output of 4.2 V, which would send an insignificant ( as good as nothing ) amount of current into a human body.

Human body can endure maximum 15 mA of current. so, careful with it.

So I need a power supply for my engineering kit that is safe. I got this old charger with Input: 120V, 60Hz, 150 mA. Output: 4.2 V, 600 mA

Is this amount safe if I were to accidentally touch it?