If the polycarb around the arena fails, I'd think safety goggles would be the least of your worries. Honestly it sounds like you're aiming for sort of low-spec Featherweights - acrylic is great to work with and laser cut, but it's very brittle to be using for a Featherweight, as far as I know. How many robots were you thinking of building? If it's more than two or three, things could start getting costly. If you're just looking to get the kids enthused with robotics I'd recommend Antweights. Cheaper, less dangerous and more manageable, and that means your arena will be cheaper too. Polycarb is certainly not cheap, and having enough to enclose that arena at a thickness you'd need for Featherweights could really cost you.
If you're still set on Featherweights though, could you give more details on what components you were hoping to use? Like maybe an eBay link or something to give an idea of what you're hoping to work with. Remember if you're doing Featherweights, those drive motors will have to drag 13.6kg around.