There's nothing inherently wrong with using small fermenters.
You can sometimes still find apple juice in 1 gal glass jugs. Those make good fermenters and I used them to make tiny batches of mead back in the mid '90's. You can also find 3 gal carboys which would be acceptable for primary fermentation of ~2 gal batches. You can probably also find food grade buckets smaller than the standard ~7 gal bucket.
That said, the kits that you can buy are typically geared for 5 gal. You could take a tested recipe and scale it down to 1 gal, but you'll probably only end up mad at yourself for only having made 1 gal of great beer. The amount of work required doesn't scale with the volume very closely.
~7 gal plastic bucket fermenters are fairly cheap (get 7 gal 'cause you need headspace for a 5 gal batch).
Don't worry about your first batch. Providing you follow the directions of the kit or tested recipe, it'll almost certainly be great. Also read How to Brew
. Actually, read it, read it again, read it one more time for good measure, then ask any questions you still have here. Then brew your first batch. Then read it again a few more times.