Earth overshoot day: is it a sound concept?
TL;DR: it's a simplified measure of sustainability, but accurate enough to be useful for public engagement 

EOD is hosted and calculated by Global Footprint Network (GFN), an international think tank. The GFN estimates national and global net supply and demand for renewable resources, specifically:


food and fiber products,
livestock and fish products,
timber and other forest products,
space for urban infrastructure, and
carbon sequestration.


From the website: 


  Global Footprint Network measures a population’s demand for and
  ecosystems’ supply of resources and services. These calculations then
  serve as the foundation for calculating Earth Overshoot Day.
  
  On the supply side, a city, state, or nation’s biocapacity represents
  its biologically productive land and sea area, including forest lands,
  grazing lands, cropland, fishing grounds, and built-up land.
  
  On the demand side, the Ecological Footprint measures a population’s
  demand for plant-based food and fiber products, livestock and fish
  products, timber and other forest products, space for urban
  infrastructure, and forest to absorb its carbon dioxide emissions from
  fossil fuels.


The results are reported in 'global hectares', the area which would be required using global average productivity, to aid intercomparability.

The methodology was published in 2013 (Borucke &amp;al 2013).

So, to answer the question in your title ("Earth overshoot day: is it a sound concept?"), data quality may be an issue (I haven't pored over the paper in detail) but the basic methodology is arguably appropriate for the question they're asking. The only issue I might have is that they effectively consider land and sea use to be interchangeable, but I'm not sure the alternative would be an improvement.

To answer the second part of your question ("what are we all living on then?"), it is possible to harvest more from a system than it can sustainably produce. Exceeding this harms the future productivity of the system. To take the specific examples named for EOD:


Overharvesting plant-based food and fibre products can result in soil acidification, soil erosion, and/or soil salinification;
Overharvesting livestock and fish products can result in overgrazing and/or overfishing;
Overharvesting timber and other forest products results in deforestation;
Using more space for urban infrastructure reduces the availability of land for the other examples on this list;
Producing more carbon emissions than can be absorbed by its forest leads to an increase in atmospheric carbon, resulting in climate change.


Ultimately, while 'Earth Overshoot Day' is a public engagement activity, there is a reasonably robust and transparent methodology behind it which is good enough for the job. Remember: all models are wrong, some models are useful.