The biblical antonym to 'watchfulness' is 'sleep'.
1 Thessalonians 5:6 Therefore let us not sleep, as do others; but let us watch and be sober.
Sleep here is not head-on-pillow sleep (it goes without saying, yet when preaching to the church Christian level one has to say that which should go without saying).
How do you 'sleep' while seemingly 'wide awake', Christian?
By drifting through your average day, your average moments and events, in a state filled with uncontrolled thoughts and emotions (daydreams, fantasies, negative imagination, mechanical associations, i.e. an ongoing series of associations spurred by chance external impulse, resentment, indignation, etc.) and in a state of being not-separate from the world about you. Man-fearing is foundationally a state of being in psychological and emotional bondage to the world about you. You are concerned with other human beings and their opinions (and their opinions of you). You are also in bondage to man when you revere man. But, foundationally, it is a matter of separation (or lack of it) from the world. You are not in a state of self-watchfulness. That requires effort. You are drifting in a comfortable state of sleep. As the Bible says: now it is high time to awaken out of sleep. (Rom. 13:11)
What do you think Jesus and Paul are talking about when they talk about 'watching' (especially when they couple the word with prayer and fasting)? Watchings, watchfulness, watch. It is an effort to be awake in the moment and to hold that state. It is not easy. This is why it as a practice is usually found in Scripture next to fastings and other things that one has to endure. That require effort.
You say, but the Bible doesn't say what it is! No? The Bible clearly says it is the opposite of 'sleep'. And, again, not head-on-pillow sleep.
This practice of watching is central to the practical level teaching of the Word of God (I've gone into, in other posts, what it actually accomplishes, but I won't here now). It's totally ignored by the church level. Why? Because the church level by definition is a level where sleep and comfort is made holy. Eat your ice-cream, go to church, talk about your kids, read books about the Bible rather than the Bible itself. Lay the foundation of the faith endlessly over and over without ever building the structure on the foundation which requires effort. Be comfortable. And asleep. Anyone who shows up not asleep is attacked by the church level, man-fearing fools like white blood cells attack disease. Only the body in this case is the disease.
So, you ask, how do you actually practice 'watchfulness'? Just try it, Christian. It's scary territory for an innocent pilgrim that's never left home before, but if you are going to call yourself a follower of Christ you need to make the move away from the world at some point. (It is a vertical movement, Christian; and it gets you noticed in the spiritual realms...)
There is an Old Testament phrase used often by men who come into the presence of God: I am here. Or, here I am, Lord. These are men who are in a heightened state of presence. A 'not-normal' level of awakeness. Actually as if they are in the presence of God Himself.
Imagine you are in the presence of God. Standing in the Court of His Kingdom, before His Throne. You will be more awake, no? Your mind will be awake to your immediate surroundings and you will be circumspect and your thoughts will be controlled and focused. Even your body language will reflect a higher state of presence and control and awakeness.
You will by default have a higher perspective on the things of the world. What may concern you greatly when in the world and asleep ("David and Amy said that movie was good, but it wasn't, and my opinion is never taken seriously by anyone...") is small and insignificant compared to being in the presence of God.
Of course this will cause you to be separated out from the world. This is not pleasant for an innocent pilgrim not used to being separated out from the world. It is very unpleasant. Very, very unpleasant. What happens is you make yourself a target. The world, the devil, and your own carnal self attack you. You put yourself on the battleground. The battleground a Christian is supposed to be on.
It's on this battleground that you provoke your limits; and it is also only on this battleground that you are able to extend your limits. To glorify God when in the midst of battle is to extend your limits. (The usual result, in the beginning, is anything other than glorifying God; but you have to see that via direct experience.)
Watchfulness as a practice is measured simply by time. 40 days; an hour; a night. A few moments is difficult. The disciples kept falling asleep when Jesus asked them to watch for just a little while.
Being in a state of watchfulness for any length of time is being separate from the world. Can you handle it, Christian? Do you want to handle it? It is a new world. To be separate from the world is to bring friction upon yourself. But this is the territory you have to get yourself on to develop as a Christian.
Try it. See how difficult it is to get into and to hold a state of heightened awareness of your surroundings and yourself -- as if you are in the presence of God Himself. Control your thoughts, control your body language, get the feel of having a higher perspective feeling in the moment; and hold it. Try to hold it during the course of a short walk, for instance (ultimately you want to be able to do it in all the traffic of everyday life, but there is also biblical warrant for doing it, as Jesus, off on your own). You'll have to actually say to yourself: I am here. And then look around you at your surroundings and get a sense of being present in your surroundings. It's a double attention: you are seeing your surroundings while at the same time you are being aware of yourself present in your surroundings. This is presence. Use the example of being in God's presence. Common, small, everyday thoughts will not possess your mind if you are in God's presence. If you have trouble getting out of common, everyday thoughts just think of something more important. Your own inevitable death can do it. The immensity of God's creation above you and around you. Thinking of central truths the Bible contains can do it.
You have to be conscious of your body and five senses as well. Feel yourself in your body. Each part of your body.
It's as simple as: "Here I am." Look around you. "I am here." See how long you can hold that state of presence. The state itself is hard enough to hold, but later you combine it with prayer and fasting, but for now the state itself is enough to give you an experience of the difference between watchfulness and sleep.
Because inevitably you will fall back into sleep. It is inevitable. Your limit for maintaining watchfulness is small, currently. Then when you, later in the day or week, remember the experience you can look at the intervening time and know - by experience - what sleep is. You will say: "Man, I was just 'gone'." I.e., you were 'gone' when you inevitably fell back into waking sleep after having tried to be in a state of awakeness. You can only 'see' sleep by seeing it in contrast to an intentional effort to be awake. That is sleep. Sleep in the world. In bondage to the impressions and influences and illusions of the world.
No, your pastor or seminary professor or Christian book writer never told you about this practice. You want to call it 'new age'. Call Jesus and Paul new age, man-fearing innocent. Or, develop in the faith, and don't fear the battefield. Don't fear being separate from the world. Fear only God, and know you are not of the world. Then actually live that and in the process build yourself up in the faith.