Soviet Russians?

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Do you mean just ethnic russians? You're aware that there were ~300 ethnicities in USSR? Quite a number of non-russians could be counted among 'most famous' in USSR. In that, case, you should ask about 'famous soviet people'.

OK then, limiting to ethnic russians and off the cuff - (and I'm limiting myself by 5 min)

Yury Vlasov - heavyweight weightlifter, widely known athlete in 60-70s, then turned to writing and politics

Kuzma Petrov-Vodkin - painter

Sergei Prokofiev - musician

Maya Plisetskaya - most famous ballet dancer from Bolshoy.

Mikhail Bulgakov - doctor, journalist, writer and intellectual.

Vasily Shukshin - writer, actor, intellectual. Hugely popular in 60s-70s.

Vladimir Vysotky - actor, songwriter, singer

Evgeny Vuchetich - architect, sculptor (oh wait, he is half-serbian)

Hmm - 'famous businessmen' in USSR - officially none. Private ownership of production means was officially forbidden, so you're either famous or you a businessman on a black market, praying not to become famous to law enforcement. Also, black market 'famous operators' and criminal masterminds were often one and a same person - see Vyacheslav Ivan'kov aka Yaponchik, for example - he made a lot of money by managing underground production of textile production knock-offs from western brands and selling it on flea markets. Alexei Kosygin would be an example of effective 'official' manager of industrial capacities.

Nitpicking aside, Artem Tarasov was first official soviet millionaire, self-declared it in 1989 on TV and quite famous. Counts as crazy and businessman, right?

time is up...

Sources

1. RT : list of prominent russians. Note that list is not divided by ethnicity and not limited by USSR. Many non-ethnic russians are listed.

2. Wiki - list of russian people. Some categories are divided into Imperial, Soviet, Post-Soviet periods. Again, many technically non-russians are counted as russians - that's typical not to separate by ethnicity (well, most of times and by most of sane ppl).

3. Book serie Life of Remarkable People = Жизнь Замечательных Людей (ЖЗЛ). Contains many books, dedicated to prominent russians of USSR. Not all books of high quality, but many are well-researched monographs.