I realize this is an old (ancient...) post, but I found it on google, and I joined your forums, so shoot me.
'Cheap' vodka is obvious, but drinkable is relative. Some people can stand to take shots of 5 O'Clock (less than $15 for a half-gallon around here), while some people can't stand Skyy even when it's mixed (I like Skyy, personally...) but here's the thing - and I'm amazed that more people haven't heard of this - filter it.
I'm sure many of you have read vodka labels where they've said "charcoal filtered" or "triple filtered" or something of that sort. Don't think they're doing some super-scientific expensive process - because they're not. In fact, it's super-ultra simple and easy. It's weird, but I heard about this on TechTV (which is now G4, and unwatchable) during the show "The Screen Savers." (it's somewhat possible that the episode is still indexed on the site somewhere, but since G4 took over, I want nothing to do with them, or their website).
Anyways, here's the deal - there's two options here, one is a little more expensive, but technically easier, and the other is cheaper but involves a little more work (go figure...).
Option 1: Britta.
Buy a Britta water filter, the kind that's a jug that you fill a reservoir at the top, and the liquid goes through the filter and collects in the pitcher. Among a few other minor ingredients, the major ingredient in the filter is activated charcoal. Many "top shelf" vodka producers filter their vodka through activated charcoal - once, twice, or more. The Britta filter isn't designed for this, so it's not as efficient at filtering alcohol as it is water.
So instead of filtering it once, filter it about 3-4 times. Usually my friends and I (generally before a party... if it's just us, we'll just deal with the awful vodka

) filter our vodka about 4 times if we're using a Britta filter. Just keep the container the vodka came in (we'll usually get 5 O'Clock when doing this, oh so cheap), and filter it through the Britta, pour it back into the bottle when it's all done (the Britta filter we got holds a gallon, but the reservoir doesn't hold nearly as much), and pour it back into the reservoir and repeat. For your first time doing this, if you're skeptical, try a shot after each time you filter (you have to try one when it's in its putrid, unfiltered form too, for science's sake). After 3 or 4 filters (depending on the new-ness of the filter you've got in your Britta thing, and the nastiness of the vodka to begin with) you'll have very smooth vodka, with very little aftertaste. The burn is still there, but that's the alcohol, and unavoidable - but even then, the burn isn't as bad, and goes away more quickly. We've used one Britta filter for two gallons of vodka, and it still works pretty well. Don't expect it to last as long as when filtering water though.
Option 2: pure activated charcoal
This is cheaper. To find activated charcoal, visit your local Walmart/Meijer (I'm in the mid-west) etc. and find their animal supplies. If they've got a freshwater fish section, you're made. (I don't suggest pet stores as much, because they often have a higher price, but if all else fails, you can always find it at a pet store that carries fish.) There are probably other places you can find it, but I'm going with what I know

.
Use a coffee filter, and fold it Chemistry-class-wise (a google search may turn something up... basically you fold it in half, then fold it in half again, then open up one of the cones [there should be two]). Fill it with some activated charcoal (more = more filtration, and it will last a little longer [ie... maybe two or three times through], but if you've got a cheap source of a. charcoal, your best results would be from changing it every time), and pour some water through it until it doesn't run grey (there's always some dust that will get through until it is washed out. If you skip that and just put the alcohol through, it will turn charcoal colored, and have some a. charcoal in it. It won't hurt anyone, but it's not as good for looks.). After it's clear, start filtering your vodka. Best results would be from changing the filter and a. charcoal after each time through, but a half-cup of a. charcoal is usually okay at filtering two or three times. Using this method, depending on how much a. charcoal you're using, and how many filtrations you let it go through with one filter, we usually say it's good enough after about 4 times through.
Presto-Chango : Top-shelf vodka.
So, my advice is to go buy a half-gallon of paint-thinner/rubbing alcohol vodka, and filter to your heart's content.