I agree wire would probably be the cheapest. It only needs to be a couple of feet high. In my experience rabbits will go under or through a fence, not over one that is a couple feet high. That way you could just step over the fence and not need a gate, which is pretty expensive. Utility wire will last longer than chicken wire.

I used chicken wire which Lasted 34 years until I moved. It became rusty but kept the rabbits out. I make mine 5 ft. tall and folded 6 inches under ground to keep critters from digging under.

I fashioned a gate stapling chicken wire to a frame made with 2xs2's. I used 10 foot 1/2 inch pieces of conduit cut in half for the stakes. All in all very inexpensive. For deer I ran fishing line across the top at 6 and 7 feet on longer stakes at each corner.

I agree that pallets would offer too much shade.

Deer? None of these solutions would work. Keeping deer out is not going to be cheap. Maybe get a dog? That would work well for rabbits, etc. as well. A deer can jump over a single pallet easily.

We used metal fence posts and chicken wire, worked good. That was the least expensive option we could come up with that would work. However, after the season was over, I took some stuff over to our local "Habitat for Humanity" resale shop, and saw lots of fence posts there, along with some lumber and fencing materials as some have mentioned. So you might look around at some type of resale place first if you have that kind of thing in your vicinity.

Here's our garden fence, designed to keep out chipmunks, groundhogs and deer. It worked for that. Curled wobbly top discourages climbing. Raised bed made out of treated lumber, which is much cheaper than cedar. We has a roll of this impervious black plastic sheeting stuff left in our garage from the home's previous owner. We don't use that kind of stuff normally, but we wrapped the inside of the treated lumber with it to discourage any leaching. Modern treated lumber theoretically doesn't leach that much but we had the stuff to use so it allowed us to save some money on the raised beds. Plus cedar boards are hard to come by where I live. We went with raised beds due to it being along the driveway and we didn't want dirt washing onto the driveway, among other issues. You might also check out some building demolition areas, sometimes I've gotten old bricks for lining garden beds in such places.

Deer don't have any trouble at all jumping over a five-foot fence BUT, if the enclosed area is very small, a deer will be smart enough not to try it, because there is not enough landing space. Not clear that 4x8 would be small enough. I know people who put up fences of that size around large beds, and the deer just ignored them, and feasted on the contents.

My community garden had a flimsy 6' fence enclosing 20-odd plots. Many people had talll tomato cages and other structures, giving the deer no place to land. The gardens were safe until fall when too many people cleaned up, removing the structures. In leapt the deer, chowing down on the veggies of those hoping keep the season going a little longer.

I saw a deer impale itself once trying to jump a taller, sturdier metal fence with a spikey top. It was horrific. They are insidious/relentless. So are groundhogs, which we constantly battle.

Metal posts and wire. Pallets will work fine if they are at least 2 feet away from the edge of the garden, 3 feet would be better on the south side. Then they would block only a little light early or late in the day. I have had good success keeping rabbits out with only 16" of chicken wire around my wood yard fence. Yes they could get in but out of sight out of mind. We have 3 dogs. Once a bunny gets in they keep coming but it keeps 95% of them out.