you know, the titanic figures, such as Aristotle, Plato, the Socratic Dialogues, St Augustine, St Anselm, St Thomas Aquinas, Descartes, Voltaire, Kant, Hegel...
...and if access was not legal but rather easy regardless, how did that system work? I remember the thread about how Western pop music found its way into the Soviet bloc, and if the purveyors were prosecuted they would be prosecuted under the don't-invest-and-make-a-profit type laws...
Most of these major philosophical figures would have been available in translation. Soviet censors during and after the Thaw tended to be preoccupied with contemporary works, so classic literature in the Western canon such as Dickens generally got a pass, and this included classical philosophy. The translations were under the auspices of state institutions, frequently in large multi-volume sets, and featured commentary by various Soviet specialists to frame each work.
How a typical Muscovite or other Soviet citizen would run into classical philosophy is another matter. Certainly there were self-made intellectuals within the Soviet Union who cultivated their own education. However, many Soviets citizens would have encountered classical philosophy within a classroom (as in the West). Here, the Soviet pedagogical approach to philosophy splintered along several lines. Some teachers would emphasize each philosopher as reflective of their respective socioeconomic epoch. Another common approach was to teach the history of philosophy as a teleological process leading to the discipline's culmination into Marxist-Leninism. Still other teachers would reduce philosophical debates into a struggle between idealists and materialists.
Sources
Ballestrem, Karl G. "Soviet historiography of philosophy." Studies in Soviet Thought 3, no. 2 (1963): 107-120.
Zweerde, Evert. "Recent developments in Soviet historiography of philosophy." Studies in East European Thought 39, no. 1 (1990): 1-53.
_. Soviet historiography of philosophy: istoriko-filosofskaja nauka. Vol. 57. Springer Science & Business Media, 1997.
As /u/kieslowskifan wrote most of classic western philosophy was available. Some eastern philosophy was also available, such as Tao Te Ching by LaoZi.
Also, philosophy was a required course in graduate schools (at least in eighties). Without passing philosophy you could not get PH.D. diploma regardless your specialization. Half of the course was classic philosophy, and the other half was "modern" one, mostly Marxist-Leninism.
Ironically many of early Marx' works were restricted and were only available to the "professionals" through special permissions.