If the chickens will be safely locked indoors by nightfall every day of the year, personally I think 2x4 fencing is perfectly reasonable and sensible, particularly if you add something smaller-mesh along the bottom 2-3' to prevent reach-through.
There *are* a variety of critters that can go thru that size mesh (I mean, their whole bodies can go thru so they'll enter your run): but sparrows and rats are hard to keep out no matter *what*, and baby raccoons and baby possums are only a very rare and nighttime threat (at least, I've totally never seen *baby* coons out during the day, only adults); and weasels and mink are pretty exclusively nocturnal hunters.
While hardwarecloth is still probably the best material for preventing reach-through, personally i think that chickenwire or 1/2" plastic mesh are not all that bad either for this specific purpose.  Reason being, the main issue is just a) to keep chickens from sticking their heads outside the run, and b) to slow things down so that if a predator DOES try to reach in and grab a handful of napping chicken, it happens slowly and awkwardly enoguh that the chicken will have a chance to clue in and get away. So if budget restricts you to a less-secure form of smaller mesh stuff, I absolutely definitely think it IS still worth doing and you won't be THAT much worse off for it not being hardwarecloth, IMO.
Good luck, have fun,
Pat