If you want to learn veneering to increase your skillset, then go ahead. If you are doing it purely to save money, I think that you will be disappointed. I had the same thought some years ago. I have done some hammer veneering. Not a huge amount, so I offer no advice on technique.
Firstly, you need something to veneer onto. This is the "substrate". You need something that is stable, so solid timber is out, and you will need a sheet material. I have used both moisture resistant mdf (MRMDF) and ply. Birch ply, which is the most expensive ply I have ever seen, at £60 a sheet for 1/2". MRMDF is cheaper, but sometimes the colour shows through. It did on some rippled sycamore veneer that I used, and gave it a green hue. I have seen threads from the experts talking about MRMDF sandwiched between very thin ply, but that is beyond my knowledge.
Wherever you need to cut the box, you expose some edges. you can either veneer the cut edges, which I really struggled with when I practiced, or you can include some strategically placed solid pieces. I know these as solid wood clippings, but that may not be the correct term. If you do that, you need a very small amount of similarly coloured timber. Buying such small amounts is expensive and difficult. You will need these clippings where the box opens, probably at the top and the base. Your 4" high box probably has 3 1/2" pieces of solid in. True, the sheet material will last you a while, but you have to store it so that it is flat.
You will need a lid and a base. I used a thinner piece for the base. I think it was 1/4", and probably another £20. That size is handy for making bases for insert trays, so I like to have a bit on hand, but it is all expense.
you will need to buy veneer. Probably small quantities, and you will want nice stuff, so that will cost. It often isn't terribly expensive on ebay (offcuts are often well sized for boxes), but postage costs may well be a big % of the order. Again, if you buy more, you have to store it and look after it. You may need to flatten the stuff, although I have been pretty lucky in that respect.
For hammer veneering, the tools are fairly minimal, but you will need a veneer hammer (or make one), an iron (don't use the household one!), a scalpel, possibly a glue pot.
I don't want to put you off. it may be a technique that you partially use- I tend to use it more often for box lids, rather than sides. but I don't want you to just get a veneer hammer and think that the expense is done. I would do a few more in solid wood first.
To keep the cost down, look out for offcuts. This is an example, made by Pete Maddex. The lid is from a piece salvaged from Custard's woodturner pile.
oak-knot-box-t100647.html?hilit=box%20custard. There are a range of cheaper timbers around, oak, ash, cherry, maple, others besides. laying out £20 on one of these will buy you enough to make a couple of boxes.