Traditionally, carbonation in either wine or beer is there from the fermentation of sugars into alcohol by yeast. The carbonation is CO2 (carbon dioxide) which is a waste product of yeast during this bio-chemical process. When you have a fermenting batch of wine or beer it will remain carbonated for the length of fermentation but if at least some of the fermentation happens in a closed container like a glass bottle or wooden barrel then the CO2 can become trapped and enter solution in the beverage. At first the additional fermentation in a barrel would have likely been accidental but eventually a secondary fermentation would be purposely started in a cask by adding additional sugar. Still the carbonation would be less than you would used to because wooden barrels can not withstand the pressure bottles or kegs can.

So with the exception of those drinking home brew that is still fermenting it wasn't until the celts invented barrels sometime after the death of christ that stored beer would be carbonated and even then it would be variable and less than you are used to now.