Restaurants absolutely existed in a more or less recognizable fashion.

Food and the City in Europe since 1800 has a chapter about Soviet restaurants in the 1930s. Its important to realize that by the 1930s a kind of consumer culture emerged that was state approved - after all socialism was supposed to provide abundance (even if we know that ultimately the Soviet economy was very frequently plagued by a shortage of consumer goods). So, restaurants, cafes, bakeries, and the sort were fairly common place and in 1935 the Soviet government put into place new standards for the quality thereof.

This is tied very much to something close to my own research, which is the emergence of a very keen interest on the part of the Soviet government in kultur'nost' or "culturedness." If socialism was supposed to be creating an "equal" world, it was also supposed to be creating a, for lack of a better term, high level of culture and creativity in the people living there. Of course, I say this full well with the acknowledgement that a "high level of culture" is entirely subjective and is not universal. But on the other hand, to the Soviet state the attitude might be summed up with the term "Soviet champagne" (this is another thing mentioned in the aforementioned chapter, in fact, but has been mentioned by many other scholars). These policies were sometimes contradictory, as you might imagine. There was a great deal of disagreement over what exactly constituted "culturedness" in the context of a socialist country was not always clear.

The highest quality things, of course, were not always available to the average person. Also, the centrally controlled economy also meant that the state controlled the opening and closing of restaurants. An interesting anecdote was the Hotel Moskva - which was so luxurious and expensive that the year it opened (1940) that historian Jukka Gronow's research found that not a single other cafe was opened that year in the Soviet Union. More average restaurants and "canteens" existed as well and were often run by factories or other employers (again, state run).

Things were paid for with money, as you'd expect. Rubles (and kopeks) were the currency in the Soviet Union - and still are in Russia for that matter - and people would spend their cash the way you'd expect. This doesn't have much to do with restaurants, but it is also worth noting that with shortage in the economy, often times cash became less important than who you knew and what favors you could trade (see also, the term blat that describes this kind of informal "currency"). This sort of thing could be helpful in securing luxury goods (or even every day goods), and although I can't think of any specific examples off hand, I'm sure it helped people get into restaurants, get meals, etc.