In popular literature the word "balance" usually means you must balance work and rest, reason and feeling, masculine and feminine. It sounds nice and explains nothing. Balance is a regime: a mode the system enters, keeps falling out of and keeps returning to. You cannot stop inside it; you can only live in it.
The engineering name for that is feedback.
Homeostasis: engineering inside the body
In 1865 Claude Bernard stated it clearly: the internal environment of the organism stays constant despite change outside. Today we know the numbers. Body temperature holds near 37 °C, give or take half a degree; blood pH stays between 7.35 and 7.45; glucose keeps to a corridor of 4–6 mmol/L.
The body does not "find balance" and then stand in it. Those numbers are held by dozens of feedback loops that measure deviations and correct them all the time: when temperature drops, shivering and vasoconstriction switch on, when it rises, sweating and vasodilation; when glucose climbs, insulin is released, when it falls, glucagon.
What you are looking at is a system with a regulator, with a setpoint, sensors and effectors, and "opposites in harmony" has nothing to do with it. When regulation breaks down, as in diabetes, no "inner wisdom" is held in reserve: insulin has to come from outside and blood sugar has to be watched from outside too.
Cybernetics: the same for any system
Norbert Wiener (MIT, Cybernetics, 1948) showed that any system with negative feedback behaves the same way: it measures the deviation, applies a correction and returns to the setpoint. The same mathematics describes a thermostat, a heartbeat, a market price, and a political opposition acting as a stabiliser of power.
Balance, then, is described by the dynamics of return. The state of a system is set by how often it returns, how far it strays and how long it takes to react.
Feedback that is too fast makes a system oscillate, like a crane swinging on its cable; feedback that is too slow lets it drift, like a neglected political regime. Everything interesting happens in the narrow band between those extremes.
What it means for a human
Two things follow.
First, balance means return. A system that never deviates is dead, because it has nothing to stabilise. Life is, by definition, a sequence of deviations and returns, and stability is measured by how well oscillations are damped.
Second, different systems regulate at different speeds. Body temperature is regulated over seconds, body weight over months, psychological state over anything from minutes to years, and confusing these speeds is the source of most mistakes. Trying to fix depression in a day works about as well as heating a room with a thermometer.
Where the evidence runs out
In biology, homeostasis is a measurable engineering property. In psychology, "emotional balance" has neurological correlates (the prefrontal cortex modulates the limbic system), but a precise cybernetic model is only beginning to take shape. In social systems it is murkier still.
The principle is the same everywhere: feedback with the right time constant. Where it exists, there is stability; where it does not, sooner or later there is catastrophe.
"Balance of opposites" is a poetic way of saying "a stable regime with well-tuned feedback loops". The poetic version is easier to remember. The engineering version works.