You don't generally dig holes for ground rods - they are intended to be driven (hammered) into place, and make better contact with the soil that way than if placed in a dug hole.
In addition, a shorter rod will be more prone to having the soil around it dry out .vs. a longer rod, leading to higher resistance to soil. Ground rod length is mostly about reaching damp soil.
Mind you, code only calls for them to be below 3 feet and is perfectly happy if they are laid horizontal in a 3 foot deep trench (actually, that now is only the case if there is rock preventing them from being driven deeper.) However, code also calls for them to be a minimum of 8 feet long, so 5 foot rods do not qualify as meeting current code (in the USA.) But for best effect and contact with moist soil, deeper is better, and exceeding code minimums is perfectly acceptable.
You can dig a bit of a posthole (2-3 feet) to make it easier to get started driving the rod. If you dig a trench, you can make it easier to start driving the rod and actually finish driving it down near the bottom of the trench, so the whole rod (and ground cable) is deeper than it would be if driven from the surface. You need to use connectors rated for burial, but most ground rod clamps are.
Multiple ground rods (spaced 6 or more feet apart) can help reduce total resistance, but the individual rods can't be too short.