Surprised to see this wasn't in the FAQ, I hope this question isn't too simple or "dumb".
Small businesses like barbershops or grocery stores were obviously necessary in the USSR, like they were in any developed nation at the time. How were these started? Could any Soviet worker decide to start a barbershop? Did the state decide where they were placed and how they were run?
Depends on the specific era you're talking about.
Immediately following the Revolution, during the Civil War, the Bolsheviks instituted what's usually called "War Communism" - state control of everything necessary to defeat the Whites. As far as smaller businesses go, I'm not entirely sure.
In 1921, the Bolsheviks instituted what was called the New Economic Policy (NEP). See, the Russian Civil War was utterly devastating, with more than 10 million people killed, and that followed hot on the heels of World War I with its millions dead. More than just the number of people killed, infrastructure was destroyed and the Soviet economy was in complete disarray. Peasants went on strike in protest of grain requisitioning and the lack of democratic representation, and what really scared the Party was the mutiny of the Kronstadt naval garrison (Although Trotski put that down rather bloodily). NEP was mostly intended as a way to get the peasants to get back to work and quiet down. Requisitions were replaced with a tax in grain, and remaining crops were permitted to be sold in village markets. Smaller private businesses were also permitted in the cities, with a limit on the number of employees. All heavy industry and foreign trade remained in the hands of the state.
Whether NEP would be continued was a major question after Lenin's death and before Stalin solidified his power. Trotski was opposed, Zinoviev and Bukharin were in favor of maintaining it. Stalin was allied with Bukharin at this time, so he ended up maintaining the NEP for a while.
It was a successful program, but the folks running the country wanted to industrialize more quickly. In 1928, Stalin began to end the NEP. Local party leaders were ordered to give tough punishments to "grain hoarders". Private farms were collectivized, and successful private farmers (Kulaks) were first heavily taxed, then later exiled to Siberia. Successful small business owners (Nepmen) were similarly punished. This is the start of the first five-year plan, wherein the state would seize control of every aspect of the economy in the hopes of forcing widespread industrialization.
The Soviet economy would remain state-controlled, down to every little business, until Perestroika in the late 80's.
Sources: Stalin: A Biography by Robert Service, with some supplementation by my notes from the Soviet History classes of Professor Steven Miner, Ohio University.