What you're proposing is basically a sort of anarcho-capitalism. Without funding via taxation there simply can't be government in the traditional sense, and you'd have private corporations instead.
However this idea is fraught with dangers because corporations don't have a vested interest in regulating themselves or benefitting consumers. They exist only to maximise profit. A police force and legal system which exists for profit is kind of terrifying.
Many however would argue that markets and corporations do self-regulate effectively, but this argument is based in faith and not evidence. That is the opinion of the Bank of England and its Governor Mark Carney. His September 2015 speech "Three Truths for Finance" describes three lies commonly believed of modern finance: this time is different, markets always clear, and markets are moral.
"Beneath the new era thinking of the Great Moderation lay a
deep-seated faith in the wisdom of markets. Policymakers were
captured by the myth that finance can regulate and correct itself
spontaneously. They retreated too much from the regulatory and
supervisory roles necessary to ensure stability"
That applies however to a high tech civilisation, and to those developed nations who have diverse economies. In comparison, Saudi Arabian citizens pay no tax, all education and healthcare is free, and even receive a grant towards their first house when they marry. But Saudi is an oil rich economy, and their state oil company Aramco allows the government to afford a generous welfare state, on top of basic government expenses. So if the economy of this place is resource rich, then the state has enough revenue via exports to pay for itself and doesn't need to tax its people at all.
I would also caution that primitive weaponry doesn't discourage war; which was frequent in the ancient world because plunder was profitable. Each urban area was a source of resources and slaves, which could be stolen if you had more soldiers. The fact they had spears and bows was besides the point. Indeed conquest was still profitable until relatively recently, see European imperialism. Which was another way of extracting free labour and resources. Nowadays the cost of war is high, and the rewards low. America couldn't extract much, if anything, of value from Afghanistan after invading; but it cost a lot of money to send soldiers out there and keep them out there.
The ancient world also has plenty of examples of relatively low tax systems, largely because tax was rarely necessary. The economy was almost entirely agrarian, and almost everyone was a subsistence farmer who could provide for themselves (minus famines). The idea of income tax as we understand it is a very modern invention; it came about when Britain needed to pay for the Napoleonic Wars, and stayed around thereafter because reasons. And that's how taxation generally had been for centuries prior; required only in exceptional circumstances, like major wars.
Other systems, like the Incan empire, required taxation as a form of social insurance, which back then simply meant providing the resources to build roads and storehouses. And people were generally happy to accept Incan control because they provided empire-wide infrastructure and insurance against famine: everyone benefitted.
So the only real examples we have of taxless systems are via resource rich economies, anarcho-capitalism exists in a realm of fantasy for now. Either your nation needs to sell something which is in high demand and rarely occurring... or you need to be more creative in your approach, bearing in mind obvious problems that occur from a lack of taxation/government services.
On the other hand if the people can afford to provide for themselves (there's enough land for everyone to grow their own food, make their own house, and enough children to provide care for them as elderly, etc) then taxation could be low or public services inexpensive; to provide an Incan insurance, let's call it, of storehouses and infrastructure.