How far can mains voltage arc in air?
The breakdown voltage of air varies significantly due to changes in humidity, pressure, and temperature.  However, a rough guide is that it takes 1 kV per millimeter.

Since that's about where arcs happen, you want to be nowhere near that in a real circuit.  On a circuit board, you also have to consider conduction along the surface.  This is why you often see talk of clearance and creapage in the same discussion.

Clearance is the straightest path between two conductors.  This is where the rough guide of 1 kV/mm for arcing applies.

Creapage is the shortest distance between to conductors along a surface.  The breakdown gradient for creapage is lower than for clearance since dirt can accumulate on surfaces.  Some dirt is partially conductive on its own, but many things can provide leakage paths after soaking up some humidity.  Take a look at specs for medical power supplies, for example, and you'll see large minimum creapage requirements to guarantee low leakage currents.

There are various safety standards out there that require minimum clearance and creapage distances according to application, voltage, and sometimes environmental parameters.  For most ordinary consumer equipment, 5 mm clearance is good enough isolation between user-touchable parts and 120 V AC power.  However, you really should look at the relevant standards, especially if you are doing something out of the ordinary.