Documentary footage of the battle
Hah, hilarious, but for the interested here is a great post from ask historians about it.
You are referring to a scene from Enemy at the Gates, where fresh troops being sent to Stalingrad are only issued arms as they disembark.
Fact. In 1942 alone, the Izveshk and Tula arsenals produced over 3,000,000 Model 1891/30 pattern Mosin Nagant rifles. They produced another nearly 2 million in 1943. (http://mosinnagant.net/USSR/Soviet-M9130.asp). Another 687,426 Model 1938 Mosin Nagant carbines were produced in 1942, while 1943 production was 978,297 (The Mosin Nagant Rifle, by Terrance W Lapin).
Meanwhile by the spring of 1942, production of the PPSH-41 submachine gun was at over 3,000 units per day.
It is estimated that at it's peak, the Red Army numbered 12.5 million, and not all of them were infantry or would have carried rifles or submachine guns.
So looking at the figures we can see in 1942 alone, the Soviet Union produced roughly 4 million rifles, carbines or submachine guns, plus a bit over another quarter million semi automatic SVT-40 rifles.
Then we had another 5 million some odd Mosin Nagant rifles and carbines made just in the years of 1939-1941. This doesn't count the millions of rifles already in inventory for the Red Army, nor any submachine guns, nor any of the US military aid given to the USSR.
In a word, the idea of unarmed Red Army infantrymen and women going into battle is absurd. In addition there were well over a million assorted M1895 Nagant and TT-33 pattern pistols available for officers, tankers, pilots, etc.
Not counting Mosin Nagant rifles already in inventory at the time of the war (plus there several hundred thousand Winchester Model 1895 lever action rifles still in Soviet hands chambered in 7.62x54r, which while I am not aware of any being used in front line combat during WWII certainly would have taken a Mosin Nagant out of a rear guard soldier's hand and allowed it to go to the front) there is no reason to seriously believe that soldiers being sent to Stalingrad lacked for small arms. This doesn't even begin to look at the use of captured German arms.