Is there a change over the business cycle in how much of economic activity is market-based? Out of the total production in an economy, a big part is market-based, as when you make something and sell it. But a big part is not, as when you make something and don't sell it, but exchange it or use it in some way. Examples of this are home production, cooking, washing, etc, as well as gifts to friends, social service, etc. 

More over, the fraction of production that is market-based probably changes over the business cycle.  For example, suppose that, in a recession, I agree to cook for my neighbor every other day because we are both unemployed. He cooks for me every other day. In a boom, instead I get a newly created job which pays me 50 dollars a day, and I by a 10 dollar a day meal from my neighbor who now sells me the food he cooks. The economy is only producing 50 dollars more since we used to produce two meals a day and now we are producing the same two meals a day plus 50 dollars. However, because I pay for the 10 in food, it seems the economy is producing 60 dollars more instead of just 50.

Other things that happen in recessions that don't count as output: when you are unemployed, you fix your house, you make art, etc. Maybe businesses give away more goods and services because they have the installed capacity, but not the demand...

Is there a way to measure the size of this effect?
Yes, the two figures are different because that is how the economy works in a circular flow. To calculate the size of economy (GDP), we will find the total expenditure, total income or total production. In theory, all three calculations should converge at the same figure. However, in economies they may give different figures. It is a fault of whole concept of GDP that it does not take into account any transaction for which there is not a market price available. So, no matter how much gardening you do at your home, you are not adding up anything in the economy. So, whatever mutual understanding you have with you neighbor and whatever transactions you make, it won't be considered. 
In the second part of your example, you are spending 10 dollars and your employer is spending 50 dollars so that add up to 60 dollars. This happens because of circular flow, your income is somebody else's expenditure. 

Production for which there is no price will not be counted.