Economic effects of a Commons Capital Depository
Has anything like his proposed commons capital depository ever been tried in practice?


Sweden had such experiment, but in a much lower scale than it was originally planned and of what Varoufakis is calling for. This was ultimately due to the opposition of employers/capitalists, and the election of a right-wing government. 

The idea came from the trade unions in Sweden, and a proposal was presented in 1975 by the economist Rudolf Meidner. According to this article, its goals were:


to counteract the concentration of ownership which, from a general equality perspective, must be regarded as unacceptable; 
to increase wage earners’ influence in economic life
through ownership of capital; 
to facilitate the continuation of the solidaristic wage policy by a
mechanism to deal with excess profits (which could also be seen as an element in lessening the concentration of capital)


The fund was to be accumulated by holding a proportion of companies’ profits (originally, 20 per cent before tax), which were to be issued as new shares. This was basically an expansion of equity, to be owned commonly by all workers in the economy. After a few decades, this method would leave most of the (equity-based) companies in the hands of workers. So in effect, this was ultimately a socialisation of the ownership of firms. Quite a bold program. 

In the end, a reduced version was introduced in 1984, which lasted only for seven years. The fund was privatised by a right-wing government in 1992.

You can read more about this here and here. 



More generally, there are plenty of countries with a so called Sovereign Wealth Fund (SWF). Many of these are financed with natural resources like oil (Norway) and minerals (Chile, copper), either coming from special taxation or royalties, or because firms are owned by the state. These funds tend to be invested in a multitude of assets, not only equity. As such, they are not directly attempting to own capital (as in the case of Sweden). I am not aware of any SWF that yields a citizen's dividend though. Many are used as a buffer in case of an economic crisis, or reinvested in other activities like advanced human capital (e.g. Chile).