That is not a lot of space for roots, but if you remove grass and/or weeds prior to putting down the bed the roots can grow down into the existing soil.
And of course you can always stack the 2x4s to make the bed 8 inches high.

I have run beds with 4" cinder blocks filled with compost on top of a layer of cardboard sitting on the existing sandy loam after using a weed-whacker to scalp the ground. When I built them in spring, they struggled a bit as the weather warmed, but in future seasons were indistinguishable from a bed originally filled with nearly 2' of organic matter so long as I topped them off with the same amounts of compost each time. Those I built as winter started had no issues the coming spring.

lucille has a good answer. You can build a simple box with two by fours using just eight nails, and you have a raised bed. Even such a little height of three and a half inches helps retain dug and amended soil and mulch to where you want it, define walkways, and define beds. ItâÂÂs a four-way win. One also avoids the pressure to get a zillion cubic yards of compost/manure/peat/etcetera all at once.

Also, when you have time to gather materials at leisure you avoid wasting huge amounts of money on (long term) useless stuff like perlite, peat, and pretty much anything else bagged from the home center stores.

LATER, just build another simple box exactly like the first and stack them. Scrap two by four wood pieces used as corner posts help keep them aligned and together. Now you have a seven inch bed and can improve on that whenever you get around to it, without the need to work on many needy beds all at once.

By this time you will know what steps you can take next.

Although 6-8" is more ideal for the root zone but I've seen many plants with pretty shallow roots. Of course, that could also explain why they didn't do as well as expected.

But 4" for the root zone and then letting the plant go deeper into your native soil should be OK this year. Next year, you can consider to go with higher walls if you want (well you could do them this year also if you want to get more wood or if you were given more wood)

How about onions and radishes? Shallots, things like that.

I built a box last year out of some old 2X4's. I just dug down into the earth below and mixed it with some vermiculite and organic garden soil. When I went to pull the old plants out during the winter I was surprised to see the roots didn't really even go down much further than the 4 inches anyway. The garden also produced well, so I think you would be ok if you were able to do it that way. To be completely honest though, I did stack another 2x4 box on top of it this year to make it a total of 8 inches.

It also depends on your climate, the amount of sun the plants are receiving and how often you water. I live in Central Florida and tried using 6" of depth in full sun with a weed block fabric bottom on my box. It worked ok in the winter, but during the summer, the soil dried out too fast and the plants cooked. They also fell over a lot during thunderstorms.
But if your garden is in part shade and you water frequently, you should be fine. Make sure to add enough compost/fertilize since the plants will use up the nutrients faster in 4" of soil.

i would also add that in my experience lettuces, spinach, radish, beets, onions, all have shallow roots - but then the problem is that they are cool season veggies, so it might be fine for fall? sort of riffing on crosstitching's post - a shallow bed will dry out faster, and summer veggies need longer roots to find water and survive hot temps.

radish would be okay though? ;)