There is not a way to drive with the back open, due to the air dynamic. The exhaust is always going to come in the back. The best way to minimize this is to keep windows shut, and air vents pulling in air from the outside if you have that option on your car. This will give you a little positive pressure but not enough to really be safe from exhaust fumes. If you have wind wings on your car, not very common these days, opening those all the way so they act as a wind scope may give you enough air to keep the fumes out.
If you have a vehicle able to pull a trailer, a trailer is a good way to overcome the open hatch. Trailers are often cheap to rent. However buying a small utility trailer can be an excellent option. The nice thing about buying over renting is that trailers, have a great resale value, I bought one used for 500, and sold it five years later for 500. Buying of course depends on if you have space for the trailer and how often you need a trailer.
A roof rack may also work well. The main limitation to a roof rack is the weight. Most passenger vehicles really are not designed to work well with roof racks. However for a few pieces of plywood or half a dozen 2x4's this may meet your needs. I would not put a roof rack on my Honda civic, but maybe a larger SUV would accommodate this just fine.
Another option is to make friends with someone that owns a pickup truck. You can best decide how this works for you. I know when I had a pickup truck many years ago, I would avoid making friends with people making friends because they needed a favor involving the pickup truck, unless of course she was cute.
The last option involves plastic sheeting and duct tape. Yes duct tape and plastic the stuff of last resort when you need to temporarily seal a space. Load your stuff and then use duct tape and plastic to seal off the back end for the drive home. Might not work very well for longer drives home that involve freeway speeds. If you have an suv/station wagon you could make the seal between the cargo department and passenger department. Should not be used when wet.
Another option for a cheap roof rack is stiff Styrofoam, place the load on a couple of pieces of the Styrofoam and tie down your load by running rope through the windows. Use caution and perhaps use some kind of shield between the rope and any points that it contacts the car. Please be very careful with long lumbar like 2x4 framing studs. This kind of load is very unstable, and you will find your self with a hood full of lumbar if you need to do a quick stop.
All of these methods involve costs. If you have a vehicle that can pull a trailer buying a used trailer is by far the most cost effective, and perhaps the most convenient. Finding a bud with a pickup may be cheaper, just depends on how much gas and beer you need to purchase. Permanent roof racks with any muscle tend to be expensive, and also reduce your gas mileage over time. Plastic and duct tape do not reuse well, and also take a lot of time.
Since your just getting a couple of 8 foot 2x4's, leave the tail gate shut and run them at an angle out the passengers side window, tying them off to the mirror, depending on the size of your vehicle it is typically only a foot or two that comes out the window.