There are several points in my life where I have been a Neet and I actually found it quite enjoyable. Once I got over the feeling of being kinda a loser because, you know, we live in a society where it tells you you always have to justify your existence by being productive in the business sense, I had a great time.
I read a large number of amazing books, spent months creating interesting projects, and took long naps out in the sunshine in the middle of the day. It was sweet.
As far as getting out of being a NEET I am blessed with having my own independent money which I manage well so I honestly didn't even need a job, though I got some anyway doing various things I found interesting before becoming a stay at home parent.
Now days I work from home but honestly not that much or that hard and am enjoying my semi-Neet lifestyle. It's very neat!
I guess from an advice standpoint I would say if you find yourself in a NEET situation and are embarrassed and can't get out of it there are a couple of perspective things you can consider.
I think the embarrassment and pressures of such a situation, beyond the need for money, are mostly based on expectations and judgements, both yours and others. Like I intimated above our society is a very ambitious one and folks are judged mostly on wealth and achievement and being between jobs or training can produce a lot of stress in this regard. It can make you feel like you've failed or are stuck standing still.
Self improvement though doesn't stop at a job or training, you can spend your free time studying subjects that interest you, exercising, learning new skills, etc Rare knowledge can be quite an asset at surprising times and can round out your skill set and make you more attractive to certain employers.
Or you can tell the world to stuff their dumb opinions and go the Buddhist monk route, getting in touch with the world around you and yourself through mindfulness practice and really living your life awake and way less stressed out.
There are a whole lot of interesting things to do with your free time if you think of your time as a NEET as a blessing instead of a curse.
Finally if you're in a situation were you lose your job or finish school and have no prospects, etc. It can be good to realize that this isn't your whole life, even if it lasts awhile. It's just a moment in time and things will change for good or ill sooner then you might imagine. Enjoy being alive as best you can.