The USSR never published a GDP figure. It couldn't, because GDP makes no sense without a market.
But yes, the USSR as a whole had a third of the US's average GDP. I agree, it was less than the US. The thing that is surprising is that they got there by planning an economy with pen and paper, which is insane to me.
A central planning economy requires calculating desired outputs and required inputs through multiple layers of a supply chain. The resultant outputs through each layer of the economy, exports, and imports have a direct value that was in fact measured in the soviet currency of Ruble's - these Ruble's could be exchanged for dollars and provide a useful GDP figure in any desired currency. The basket of goods a given amount of these Ruble's could buy internally vs. their dollar equivalent elsewhere also provides a reasonable measure of Purchasing Power Parity.
One can also bypass the production component and measure GDP in terms of the incomes in Ruble's paid to Soviet workers in combination with the export/import figures to arrive at a reasonable estimate.
The basket of goods didn't have any fixed ruble price, as much of it was free.
Much of production was never given a full price in rubles, as a large part of it was by direct resource allocation.
You can indeed estimate it using a basket of goods and their equivalent values, but not with income and import/export + exchange rates because you're then removing all natural wealth.
It's a very difficult problem. You can make an estimation, but it's still an estimation, that is, for hard GDP. For PPP I agree the numbers are accurate, but that's not what was being discussed.
Something being free simply means the government paid for it on behalf of the worker. Direct resource allocation likewise just implies the government paid for it albeit without consideration for margin or potentially the ability of the resource allocator to remain solvent.
Direct resource allocation implies that valuable resources were taken from one activity and allocated to another, the workers who produce the resource still get paid (hopefully!) and the equipment still undergoes wear and tear or upgrade requirements. The production's value can still be estimated either by the equivalent market rate of the good or by the cost of making it.
When the government pays people to build housing that they give away for free there is still "production". When the government orders a mine to ship steel to another location the workers are still paid.
When the government orders a mine to produce for no input and doesn't bother to maintain equipment they effectively take a loan against the capital equipment in the mine. Eventually the mines productivity falls to zero as the mine fails to remain solvent requiring more capital infusion.
If the government can order workers to work without pay then the estimation has to account for similar economics to slave states, which unfortunately is also well trodden ground owing to economies such as the US's prior to 1865.
The issue always comes to assign a price in US dollars.
The government may pay people in other things that were the result of direct resource allocation, so you simply cannot put a price to it.
What is the cost of land that the government gives you? What is the cost of a lump of coal? Clearly this is a lot more than the simple capital cost used to extract it.
The long and short of it is, you're not getting a meaningful price is US dollars.
