Yeah the 1/4 is probably too big and you would might have to get bigger eye bolts but not real sure.
When you say you have to take it down do you mean after each time you use it or the season?
Do you have to take the cables down too or will they get to stay up permanently?
I apologize but I'm not picturing how you are going to use this. From your post it sounds like each time you use the net you will have to put the cables up and hang the net on the cables. If this is correct then I would suggest in finding another solution because that would be a lot of work just to hit.
If I'm wrong (and I hope I am) then if you are going to just slide it back and forth then you will be in great shape.
Also double check the walls and how strong the studs are because you are going to have a lot of tension on them. You might have to add onto the studs to give them enough strength to hold the cables.