Reddit mentions of LUCID Foldable Metal Platform Bed Frame and Mattress Foundation -Strong and Sturdy Support - Quiet Noise Free - Full Size

These things are the way to go! My house is tiny with a super narrow staircase so I couldn't even move in a full size box spring. Ended up messing around with pallets, etc., but the wood would splinter and pierce my mattress, or slide around and I'd wake up with my feet in the air and my head near the floor. Really the pallets got incredibly annoying.

Ended up buying something like the person above posted, except you can find ones where the "box spring" is included. That's not the exact one I own, but if you shop around, I used the keywords "collapsible bed frame box spring".

Also really convenient for moving since it comes in a giant box and just folds right up. Only took me 15 minutes to drunkenly set up.

Your budget is not bad, but, I would recommend you go down a size to full size in order to get something better.

To start out, you will need a foundation to place the mattress on.

This full size one is 85 dollars shipped: http://www.amazon.com/dp/B00ENRBD8O

On a 300 dollar budget, that's 215 left over for a good mattress that will last you 5-8 years.

For the mattress itself, at your price point, coils are always a better deal than foam. There might be places that will sell you memory foam or regular foam mattresses, but those will start wearing out much quicker than a coil mattress.

So for about 200 bucks, you can easily get a decent full size mattress.

Specs to look out for are as follows:

8" Thick Coils - Anything less than around 8" means that you're more likely to bottom out and wear out faster. Make sure to get Medium Firmness, Soft coils won't last long at this price point, Firm coils are actually about 100 bucks more. You can actually feel them through the side. Image: http://i.imgur.com/AKuEW1u.jpg

About 1/2" to 1" High Density Foam - Usually at about 2.0Lbs to 2.5" Lbs density rating. This is usually found on the mattress spec sheet, a lot of internet retailers lie about this, btw.

Done.

If you want to add additional padding, get a duvet cover.

Amazon.

I've bought this sucker twice, the second time after a fire ruined the first.

If you've got Prime it's free two day shipping. Comes all compressed and rolled up in a ~18x18xlength of mattress box and you just let it out and it decompresses into a really nice mattress.

Just make sure you have a bed frame with wood slats or a metal one like this.