Big enough for two

He probably means something like the second picture down. Unless you have a mattress specifically designed to be frameless, it's always better to have a frame. Not only will it prevent mould and morphing like he said, but it provides better support for you and lengthens the life of the mattress. If you really don't want one, it's best to instead invest in a floor mattress like a Japanese Futon, which is light enough to roll up or hang to air during the day.

Here are a couple things I've tried.

Japanese Futon

I spent a year in Japan and used a Japanese futon during that time. The mattress was about 2 inches thick and packed pretty solid with cotton. The futon was usually on a flat, but smooth wooden platform bed. I also sometimes used it on the carpeted floor if I wanted a softer surface. It was really comfortable, but you can't get away with a soft pillow. You need something that gives more support. I had a memory foam pillow at the time. I use a buckwheat hull pillow now and have a very firm memory foam mattress.

Don't get mixed up with American futon mattresses for folding frames, those run closer to 6 inches thick and are as difficult to move as a typical mattress.

Here are a few on Amazon to give you a sense of what to look for. This site also shows a three section foam pad that you can place under a futon mattress if you want something softer.

I saw "throw beds" at Target the other day for around $50. They were basically small, narrow futon mattresses. I can't seem to find them online or I'd include a link. These wouldn't be comfortable directly on a hardwood or tile floor, but if you've got a carpet or rug with some padding underneath, it feels pretty good.

I'd recommend trying something like that out on the cheap if you can before dropping the cash for a Japanese futon and possibly a foam pad.

Camping Sleep Pad

I also have an inflatable sleeping pad that's insulated for backpacking camping. It's really narrow, so it's not as comfortable as a typical bed, but the insulated aspect of it is really nice. It also packs up really small.

I slept on one of those guest bed air mattresses for about a year. I had to layer a lot of blankets under me in the winter or I felt really cold.

I got a new bed too! (https://www.amazon.com/gp/product/B012C3MZ2I/ref=oh_aui_detailpage_o00_s00?ie=UTF8&psc=1)

https://www.amazon.com/EMOOR-Japanese-Traditional-Mattress-Classe/dp/B012C3MZ2I#customerReviews

Here's my 'bed':

Futon - https://www.amazon.com/gp/product/B012C3MZ2I/ref=oh_aui_detailpage_o03_s00?ie=UTF8&psc=1

Futon Cover - https://www.amazon.com/gp/product/B00KF88DU6/ref=oh_aui_detailpage_o03_s01?ie=UTF8&psc=1

Tatami Mattress - https://www.amazon.com/gp/product/B01A9SVX0S/ref=oh_aui_detailpage_o03_s02?ie=UTF8&psc=1

I love it, and I can fold it up and make use of all the open space.

floor mattress