Raccoons can't reach through 1/2". They can climb fencing, or simply tear through it if the wire is thin enough (i.e. most chicken wire).
A small enough weasel could probably fit through 1" openings.
Depends on your predator load, predator types, and how much risk you're willing to take. If you have raccoons I would at least cover the bottom 2-3' with hardware cloth (to prevent reachthroughs), even if you opt to go for a different fencing material throughout.
Also I'd do 1/2", not 1/4"...