Is 5V 20A power supply output safe?
That power supply can, if it's demanded of it, put out enough current to start a fire in the wires or a resistive load. The wires coming from it always need to be big enough to carry 20A safely. If they are thinner, then they need to be protected by a fuse at the power supply.

5v will not give you a shock.

If the supply is mains powered, rather than battery powered, then a fault in it could give you a shock. Commercial electronic equipment is designed so that a single fault anywhere should not give you a shock, two or more faults are required. One way is that supply is grounded. If this is the case, then you must make sure the ground wire is always connected using a 3 pin socket. If the supply only has two leads, then it is 'double insulated', and does not need earthing.