Containing kittens and mother: What should I use as they grow?

There is probably a topic on this somewhere that I didn't see, but I need some advice about this.  My semi-feral I have cared for since December and took in during early April had five kittens 12 days ago. They are currently in a box with three-inch sides under a chair and are in the storage room because I have a male (neutered) cat in the house. The kittens' eyes are open and they can hear a little so they are more active, playing with each other a little and trying to explore. One of them got out of the box and was crawling on the carpet this morning but the mother cat, who was eating at the time, didn't care. The problem is that I don't want them exploring the storage room due to safety issues with too many places to get stuck, although I've tried to block off places they could squeeze into. The mother does not want a higher box. I tried but she was distressed by this, presumably because she couldn't see them and get to them quickly. I have put them back in the lower box and today put a book under the escape route to raise it a little.

Does anyone have a suggestion of how to contain the kittens as they grow?  I was thinking of a dog exercise pen but the concern is the bars and little heads getting stuck and I thought of a baby playpen but kittens could climb the mesh. Crates won't work because I have to have an easy in for the mother and yet keep the kittens from getting out.  She also needs her separate place for food/water and litter box.  I am also going to need to bring the kittens and mother in the house in a few weeks because I live in the South and it will get hot out in the storage room. It's warm now but fortunately the storage room is all brick with a concrete floor that is still cool and there are two screened windows that could be opened a crack if I could get them unstuck but that won't be enough when it hits the 90+ weather starting in June. I've put a fan up on a table to circulate the air. So....what should I get to keep the kittens secure?

Thanks for any suggestions!

Having recently come through this myself (Daisy's kittens are now 9 weeks old) I'd suggest the best bet is a different room where they are safe, the kittens will start to explore and this is a natural process I think and also leads to them following mum to her food and to the litter tray which makes weaning easier, if this really isn't possible then yes I'd say as large a dog pen / cage as you can manage, the ones I've seen have bars that are fairly close together so I don't think it would be an issue but if you were worried you could thread cardboard through the bars to close off the gaps (I did this when I used one for my pet lizards to get some natural sun)

I was going to ask if you have a small room you can put them in.  A walk in closet or spare bathroom can work very well.

If not, yes, a large pen may be in order, but this is not ideal, especially if you plan to keep mom in there too.  Try to find one with two levels if you can so they have maximum room, especially mom.

Thanks, guys.

I had a great place for her to have the kittens and keep them for the first two or three weeks and if the weather is just warm I can keep them out there in a pen of some sort. In the house,  I don't have a spare closet and the two bathrooms have sliding pocket doors which can easily be opened by a determined cat - Henryetta or, worse, Chaucer who is actually good at it. The house is pretty open-concept with the living room and separate dining room opening directly from a foyer. The kitchen/breakfast room and den area is open as well with sliding pocket doors to shut the area off from the rest of the house. The laundry/utility room was fine for Henryetta before I moved her to the storage room to have the kittens but it would not be good for curious kittens and their mother, plus laundry must be done as well and there is a floor drain. .I have guests coming for the weekend in June as well and while the house has large rooms, there are only two bedrooms.  I had thought I would  stay in the guest room if I could get a pen for the kittens and Henryetta but it's not ideal. I really do not want to share a room with Henryetta as I'm still unsure of her actions so don't want her on the bed with me - Chaucer sleeps on the bed with me normally but, of course, would be shut out of the guest room- and she may be nervous with strangers in the house and my male cat.

The crates with two levels don't look big enough to accommodate. Is there enough room for 5 kittens and their mother with a couple of litter boxes and food/water dishes that would be required. With a pen, if I could get it to stay in place, which is another concern since the floors are hardwood and the walls were just redone, I thought it could be high enough so the mother could get in but the kittens could not get out and I could keep her food/water and litter box outside the pen. I like the idea, Mum2Daisy of the cardboard up around the bottom part to keep them from climbing.

The other option for the weekend in June is to board Chaucer and put the kittens and Henryetta in a pen in the living room, although she is still very skittish of people and my friends tend to be a bit louder than I'd like them to be and would want to play with the kittens.