If you want to use "earthen" clays, you can just let them air dry instead of then kiln-firing them. They will be more brittle without high-temp firing though.
(As mentioned, if you're serious and want to go to the trouble, you can also make your own kiln or make one "in the ground." However, you could also take your air-dried earthen clay pot to a place that will fire it for a fee--clay stores, university art departments, etc).
You can buy various regular earthen clays at art supply stores, and you can often also buy various brands of earthen clay in boxes that are often purchased by kids and just air-dried after shaping (like Mexican Art Clay) instead of being fired, in craft stores like Michaels. Those can be painted after drying and/or have a clear glaze put on them (also air-dry) and I suppose they'd be waterproof afterward at least for some amounts of time but won't be the same as kiln-firing a glaze...wouldn't drink out of them.
(Plus, there are various other kinds of homemade or store-bought "air-dry" clays based on grains/flours or wood/paper rather than dirt/minerals/earth that you could make things with too, but must be sealed to prevent later damage from humidity and sometimes bugs.)
To make the basic pot shape you can use a wheel, or you can do without one and just use "slab" or coil techniques for making vessels of various kinds/shapes.
You can also make all kinds of vessels --freestanding or over a permanent armature-- from polymer clay which is not air-dry. It must be heated to harden but since it's a plastic it can't be heated very high since at about 385 it would burn to a crisp. Here are various kinds of vessels made with polymer clay:
http://images.google.com/images?q=polymer+clay+vessel
For info on making things like that if you're interested, check out at least these pages at my polymer clay "encyclopedia" site:
http://glassattic.com/polymer/vessels.htm
http://glassattic.com/polymer/vessels_rock-alum.foil.htm
HTH,
Diane B.