In the UK, I could start my own business - either as a self-employed person, or by starting a company. With a company, the company might have £50,000 income, £5,000 cost, and pay me a £45,000 salary. In that case the company has no profit and pays no taxes, but I personally pay income tax. Or I could pay myself any salary I like, say £20,000 salary, so I pay tax on £25,000 profit and £20,000 salary. The state actually gets less money in total if I set my salary so the company makes a profit.
If I'm self employed, income minus cost is my profit and I pay taxes on that. If I don't make profit, I pay no tax. Unfortunately, I also wouldn't have any money to buy food, pay the rent, and so on and so on. I'd have the same income and pay the same taxes as someone who is unemployed.
There are "businesses" that are just run for the enjoyment of the owner and don't make profit. Rich guy buys a farm and starts breeding race horses, that kind of thing. In that case, there is zero difference between a guy breeding race horses and calling it a "business" and another guy breeding race horses and calling it a hobby. Neither makes money and neither pays taxes.