OP.......how many chickens, how big of a run?
There would be several ways of looking at this. Once you put birds in a run of almost any size, you can forget about them getting anything green, so a run is basically an expanded porch on a coop. A way to allow the birds to be outside and getting fresh air. You can make it as large or as small as you want, but 10 sf per bird is the rule of thumb standard.
But assume your budget would only allow you to cover that much run space with chicken wire and not hardware cloth or my preference, 1" x 2" or smaller welded wire. What chicken wire is good for is confining chickens to a given space. They can't get through it, but a whole host of predators can. So you go with chicken wire, invest a lot of time and money into a flock of birds, and all is well for about 4 or 5 months (with accumulated costs and expenses all along the way) and within days of you getting your first egg, a predator smashes through the chicken wire and kills them all. The odds of that happening are probably 50:50 or greater, depending on your location and how secure the perimeter is beyond the run. But at that point, a run half the size of the one you built, made of sturdy stuff that nothing could get through would then look like the better way to have spent the money.
On a limited budget, and with limited space, not only will a smaller covered run be safer, but will be easier to manage. Half the area to keep mulched with a deep layer of mulch, etc. Covered runs protect from raptors, climbing varmints, rain and sun. Basically it is a secure coop with open sides.
Or, if chicken math is a very real concern, and you want to go large in anticipation of the future, build as large as you can, go with chicken wire for now and hope for the best until you can save enough to come back later with the good stuff. But don't be shocked if you come home one day and they are all dead.