Umm... we're all on a budget. OP wanted to make it as cheap as possible. That's not saying "I've got ten bucks. Let's do a reef tank!"
A responsible pet owner is someone who works within his or her budget. To say having a budget makes you an irresponsible pet owner is a pretty far reaching statement.
So rather than vague "Yes it's possible" or "It'll never work!" how about we do a cost breakdown and figure out what falls within the budget?
jkdubs2, you have a tank. Though I'd go bigger if possible. It was a planted tank, so I hope you have better than the standard fluorescent light.
Substrate: A bag of aragonite sand will be 10-20 bucks, depending on depth desired. For such a small tank, I'd go shallow. Maybe 5-10 lbs.
Rock: You will need at least 10 lbs of LR, 15 is preferable. Figure $5-6 per lb, or petco sells aquacultured (manufactured) rock by size, and you can sometimes find a "medium" for $20 that is up to 5 lbs. Let's estimate high, and say $70 for rock.
Salt Mix: $20 will keep you salted for quite a while.
Water: Tap water is not a good choice. I use Glacier water from the grocery store. 35 cents/gallon. So, $3.50 initial, and then a couple bucks every week.
Powerheads, heater, etc: Figure on $70-75 for such a small tank.
Let's say two fish. One being a neon goby ($10-15) and the other being a watchman Goby ($15-30) to pair with a Pistol shrimp ($15-30)
Snails, hermits, etc. Figure $10 bucks.
If your lighting is better than the standard (T8), you could get some soft corals (Mushrooms, zoanthids, leathers) and even some low light LPS corals. If it's the standard, you *might* be ok with just some softies. Or go FOWLR (Fish Only With Live Rock). Figure at least $100-150 if you want corals at all.
So, Here's the breakdown so far.
Roughly $180 for equipment
$50-75 for fish and inverts
If you want corals, another $100-150.
Remember that there will be costs for water, salt, food, etc as time goes on. Is this within your budget? This is what i would call bare minimum.