I would like to make a small quantity of beer, using something like this
What are the downsides of using a bucket much bigger than the amount of beer?
The headspace won't be an issue since your beer will produce co2. I recently did 2 gallons in an eight gallon bucket and one issue I did encounter is that it felt like since the beer was so shallow in the bucket, I had a small zone from which I could rack. Otherwise, for primary, you shouldn't have a lot of problems.