As far as I know, there were no cases of the Soviets sending unarmed men into combat during ww2. Poorly armed and trained men, at times, yes, but not unarmed men.

There are some kernels of truth to this story for the Russian Imperial Army during ww1 and there could be some for the Narodnoe Opolcheniye (people's militia) units of the Soviet army during ww2.

The Russian Imperial Army was unable to equip all its drafted men with rifles early 1915 - and purchases and deliveries of British and Japanese rifles were made (remaining rifles in stock were later sold at high prices to the desperate Spanish Republic 1936-38). By December 1914, the Russian Imperial Army had roughly 6,5 million men in uniform, but only 4,5 million rifles to equip them with. At the end of 1915, they had 2 million men at the front, but only about 1,2 million rifles to equip them with. However, by that time the Russian production had picked up, reaching about 1,3 million rifles per year, and there were 2 million rifles in Russian ports or on their way from Britain and Japan.

All countries fighting in ww1 suffered huge problems trying to supply their forces and equip men raised to replace the extreme casualties suffered by the pre-war armies during the first year of the war. It took time for industry to change to war production, and the lack of nitrates (in the Central Powers due to the blockade, in the Entente due to difficulties in increasing shipment of guano) hampered production of ammunition. The British had to purchase Canadian and US rifles and took rifles from the Indian army as a stop-gap measure. However, the Russian Imperial Army was the only army to send uniformed but unarmed men into combat, expecting them to arm themselves on the battlefield late 1914 to late 1915.

During ww2, the Soviets attempted several times to use Narodnoe Opolcheniye, or people's militia units as a desperate stopgap measure when the Germans advanced deep into their territory. These were civilians organised in a sort of a home guard and at times trained decently well and equipped almost like regular troops and at other times thrown into combat barely equipped at all when there was little or no time for training and little equipment available. This was especially problematic during Autumn 1941, when much of the Soviet war industry was being moved from territory that had been or would be over-run to beyond the Ural mountains, causing a temporary slump in supply and new equipment available to the Soviets. For example, the German 6. Panzer-division, when advancing on Stalingrad as part of Fall Blau in summer 1942, encountered a Soviet unit of AA guns and attacked it, noting that the enemy was firing badly, and after running over the unit, discovered that the guns had been manned by female factory workers still in civilian clothing. On the other hand, the Narodnoe Opolcheniye units part of the 62. and 64. Army that defended Stalingrad seem to have performed well during the fighting.

It is quite possible that Narodnoe Opolcheniye units were at times equipped with a lower number of rifles than it had men and went into battle like that - they often lacked heavier support arms such as artillery, AT guns and mortars. However, I have not seen any records of systematic under-equipment of Soviet units during ww2, neither regular forces or Narodnoe Opolcheniye.

Sources:

Brusilov Offensive, by Timothy Dowling.

Race to the Front, by Kevin D Stubbs.

Stalingrad, by Anthony Beevor.

The war where the Russians were unable to give all their troops rifles was World War I, and that was for a phase in 1914-5. By 1916-7 the Russian army was so lavishly equipped in arms and ammunition that it was the quartermaster of Lenin's regime for most of the Russian Civil War on the leftovers of the 1916-7 phase.

The Russian state was a world power from the point of Ivan the Terrible's annexation of much of Siberia, which put Russia on the Pacific, made it a vast and sprawling state with no real territorial equals and fundamentally changed its conception of itself, and gave it an overland region to march anywhere it wanted if it paid the price in rubles and logistics to do so. That's a far bigger caveat than it seems, and one that tends to be left out of the discussions of Russian historical patterns of expansionism relative to other powers and what they are and are not.

It became a European power following the triumph of Peter the Great in the Great Northern War at Poltava, confirmed further by the Partitions of Poland and the victory of Tsar Alexander I over Napoleon in the 1812-14 wars where he orchestrated the destruction of the Grande Armee, and built a coalition with the rather less eager to do all this again participation of Prussia and Austria that ended in a wholesale trouncing of Napoleon.

The Russia of the Tsars was literally one of the most powerful engines of conquest in world history, which is where a lot of my Russophilia comes from due to it never being given enough credit, or the right kinds of credit, for what it did and how it did it.

The USSR's ability to recreate for a time elements of the old Tsarist state stemmed from the reality that WWII destroyed the Great Power status of European states other than the USSR and colonialism reduced China and India to minor players for a generation. The USSR was the great land power with overland borders to make its projection a lot simpler in some of the most vital areas than was the case with the United States.

The fall of the Soviet Union was largely due to internal contradictions and to the nature of the Gorbachev Reforms, which loosed the power of repression but refused to wield the iron fisted approach that other Russian rulers then and later used. It was not necessarily inevitable, though a surviving USSR would have been far from the superpower it was in the 40s, and that was a blend of relative decline, of the rise of the Internet Age, and the USSR's political system ossifying in patterns that were rigid and did not permit easy resolution short of causing the system to go boom.

Reagan gets a lot of credit for the collapse of the USSR but the Soviets mostly brought it on themselves, with the final coup de grace the agreement by the Russians, Belarusians, and Ukrainians to create three separate Eastern Slavic states rather than the vast sprawling mass of the unified Russian empire.

As far as WWII, Soviet victory was half the US Lend-Lease giving it the ability to mechanize its armies at a rate the Nazis were for geographical reasons unable to slow down, let alone stop, and half the reality that the USSR had from the 30s a modern doctrine suited to waging the war it fought and experience with building vast quantities of modern weaponry and proved able to, thanks to Lend-Lease, churn out a vast weight of weaponry and use the newly mechanized forces to wield its existing armies with far greater skill than it did in 1939-41.

NATO essentially paid for the Warsaw Pact without being able to do much to stop it, because if it wanted to stop a Soviet conquest on that scale it might have been better off fighting the European war in Europe instead of the interior of Egypt because its major belligerent force was incapable in the summer of 1942 of fighting the Germans any better than it was in the spring of 1940 in Norway and France.

Of course chauvinism and the outcome of the Cold War means that it's easier to forget that the reality of a vast Soviet empire in Europe was sealed with the failure of the Greece campaign and Gazala rather than anything else.

Sources:

When Titans Clashed: How the Red Army Stopped Hitler, by David Glantz

The Eastern Front, by Norman Stone.

Gorbachev: His life and times by William Taubman

Armageddon Averted: the Soviet Collapse 1970-2000 by Stephen Kotkin

Toward the Flame: Empire, War, and the End of Tsarist Russia by Dominic Lieven

Alexander I: The Tsar who defeated Napoleon, by Marie-Pierre Rey.

Ivan the Terrible by Robert Payne