You can bury your pet in a plastic bag to help contain any body fluids from the pet.
However there's no requirement for burying your pet in a plastic bag and all you have to do is just dig a hole big enough for the pet and then place the pet directly into the hole and cover the pet up with dirt.
Dig the hole for the pet at least 3 feet deep if possible to allow the burial of the pet deep enough to prevent other animals from digging them up.
You can bury your pet in your own yard and do not need to go to a pet cemetery or do cremation of the pet if you do not want to or cannot afford to.
All pets of mine that have died have been buried in our own yard and without the use of a plastic bag.
I just always put the pet directly into the ground.